Tamás Darvas

Chinh H Lu

Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.

Introduction

In this paper we prove the essentially optimal form of Donaldson's geodesic stability conjecture [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]. The main result is obtained via a detailed analysis of the rays associated to the space of Kähler metrics.

Suppose (X, ω) is a compact Kähler manifold with dim X = n. We consider H, the space of Kähler metrics cohomologous to ω, with its L p type Mabuchi metric structures (H, d p ), p ≥ 1 [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]. For simplicity, to describe our motivation, let us momentarily assume that X has no non-trivial holomorphic vector fields. In the recent breakthrough papers [START_REF] Chen | On the constant scalar curvature Kähler metrics, apriori estimates[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF] Chen-Cheng provided the first existence theorems of constant scalar curvature Kähler (csck) metrics inside the class H. Such metrics are minimizers of Mabuchi's Kenergy functional K : H → R [START_REF] Mabuchi | Some symplectic geometry on compact Kähler manifolds I[END_REF]. Together with [START_REF] Berman | Regularity of weak minimizers of the K-energy and applications to properness and K-stability[END_REF], the Chen-Cheng results provided a full characterization of existence of csck metrics in terms of d 1 -properness of K. As d 1properness is actually equivalent with properness in terms of Aubin's J-functional [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF], this also verified an old conjecture of Tian [START_REF] Tian | The K-energy on hypersurfaces and stability[END_REF], [START_REF] Tian | Canonical Metrics in Kähler Geometry[END_REF]Conjecture 7.12], with the precise statement appearing in [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF]Conjecture 2.8].

Energy properness is the strongest form of stability. Contrasting this is uniform Kstability, one of the weakest such conditions. When the Kähler structure is induced by an ample line bundle, this criterion was first considered by Székelyhidi [START_REF] Székelyhidi | Extremal metrics and K-stability[END_REF], and was further studied by Dervan, Berman-Boucksom-Jonsson, Boucksom-Hisamoto-Jonsson [START_REF] Dervan | Uniform stability of twisted constant scalar curvature Kähler metrics[END_REF][START_REF] Berman | A variational approach to the Yau-Tian-Donaldson conjecture[END_REF][START_REF] Boucksom | Uniform K-stability and asymptotics of energy functionals in Kähler geometry[END_REF][START_REF] Boucksom | Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs[END_REF]] and many others. The ultimate hope is that (uniform) K-stability is weak enough to be verified using computational techniques of algebraic geometry, this being the main motivation behind the Yau-Tian-Donaldson (YTD) conjecture, seeking to show that some form of K-stability is equivalent with existence of csck metrics.

In this paper we focus on Donaldson's geodesic stability conjecture [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]Conjecture 12], lying between energy properness and uniform K-stability (see Conjecture 1.7 below). This conjecture predicts that it is enough to check properness of the K-energy along the geodesic rays of H to insure existence of csck metrics. Initially, the predictions of Donaldson advocated for the use of smooth geodesic rays [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]. As we know now, the typical regularity of geodesics is merely C 1,1 [START_REF] Chen | The space of Kähler metrics[END_REF][START_REF] Blocki | The complex Monge-Ampère equation in Kähler geometry[END_REF][START_REF] Darvas | Weak geodesics in the space of Kähler metrics[END_REF][START_REF] Chu | On the C 1,1 regularity of geodesics in the space of Kähler metrics[END_REF], even when connecting smooth endpoints. Hence the present expectation is that (in its optimal form) Donaldson's geodesic stability conjecture should hold for rays that have at most two bounded derivatives. In Theorem 1.8 we essentially verify this form of the conjecture.

To carry out our plan, we first explore in depth the metric geometry of L p geodesic rays (i.e. rays running inside the d p -completions of H), a topic of independent interest. To do this, perhaps surprisingly, we need to understand uniform convexity of the L p Mabuchi geometry when p > 1, extending work of Calabi-Chen in the particular case p = 2 [START_REF] Calabi | The space of Kähler metrics[END_REF]. After exploring the metric space of L p geodesic rays, we show that such rays can always be approximated via rays of C 1, 1 potentials, with converging radial Kenergy. With slightly different formulation, the uniform L 1 geodesic stability conjecture was verified in [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF], pointing out that it is enough to test energy properness along L 1 geodesic rays to guarantee existence of csck metrics. This result, together with our approximation theorems just mentioned will yield the geodesic stability theorem for rays of C 1, 1 potentials, i.e., potentials with bounded complex Hessian.

In addition to the above, our results resolve a number of related open questions in Kähler geometry, specified in the paragraphs below. Also, in the particular case when the Kähler structure is induced by an ample line bundle, our theorems also make connection with the variational program designed to attack the uniform YTD conjecture (see [START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]). Roughly speaking, to verify the uniform YTD conjecture using our results, it is now enough to show that specific C 1, 1 geodesic rays can be approximated by geodesic rays induced by the so called test configurations of algebraic geometry [START_REF] Tian | Kähler-Einstein metrics with positive scalar curvature[END_REF][START_REF] Donaldson | Scalar curvature and stability of toric varities[END_REF] (with converging radial K-energy). On the surface this sounds simpler than approximating L 1 metric geodesic rays [13, p.2], and time will tell what role our results will play in the solution of this problem.

Uniform convexity and uniqueness of geodesic segments.

By H ω we denote the space of Kähler potentials associated to H. The metric completions of (H ω , d p ) are (E p ω , d p ), and the latter spaces are complete geodesic metric spaces for any p ≥ 1 [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]. The distinguished d p -geodesics running between the points of E p ω are called L p finite energy geodesics (or simply finite energy geodesics, or L p geodesics, if no confusion arises). These curves arise as limits of solutions to degenerate equations of complex Monge-Ampère type. We recall the basic properties of these spaces in Section 2.1.

For any p ∈ [1, ∞) it was shown in [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Theorem 1.5] that the metrics d p are "convex": if [0, 1] ∋ t → u t , v t ∈ E p are two finite energy geodesic segments then

d p (u λ , v λ ) ≤ (1 -λ)d p (u 0 , v 0 ) + λd p (u 1 , v 1 ), λ ∈ [0, 1]. ( 1
)
This property is called Buseman convexity in the metric geometry literature [58, Section 2.2], going back to [START_REF] Busemann | The geometry of geodesics[END_REF]. In the particular case p = 1, (1) was established in [START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF]Proposition 5.1], having applications to the convergence of the weak Calabi flow. In case p = 2, (1) follows from the fact that (E 2 ω , d 2 ) is a complete CAT(0) metric space, as shown in [34, Theorem 1], building on estimates of [START_REF] Calabi | The space of Kähler metrics[END_REF]Theorem 1.1].

The CAT(0) property consists of the following estimate:

if u ∈ E 2 ω and [0, 1] ∋ t → v t ∈ E 2
ω is a finite energy geodesic segment then

d 2 (u, v λ ) 2 ≤ (1 -λ)d 2 (u, v 0 ) 2 + λd 2 (u, v 1 ) 2 -λ(1 -λ)d 2 (v 0 , v 1 ) 2 , λ ∈ [0, 1]. (2)
As is well known, (2) implies (1) [START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF]Prop 2.3.2]. Unfortunately, there is very strong evidence that (2) can not hold for the d p metrics when p ∕ = 2. Indeed, when restricting to a toric Kähler manifold and toric Kähler metrics, the spaces (E p ω , d p ) are isometric to the flat L p metric spaces of convex functions defined on a convex polytope of R n [49, Section 6]. It is well known however that CAT(0) Banach spaces are in fact Hilbert spaces [START_REF] Bridson | Metric spaces of non positive curvature[END_REF], evidencing that only (E 2 , d 2 ) can be CAT(0). Despite this, in the first main result of this paper we show that adequate generalizations of the CAT(0) inequality (2) do hold for the d p metrics, in case p > 1. These can be viewed as the Kähler analogs of classical inequalities of Clarkson and Ball-Carlen-Lieb, regarding the uniform convexity of L p spaces [START_REF] Clarkson | Uniformly convex spaces[END_REF][START_REF] Ball | Sharp uniform convexity and smoothness inequalities for trace norms[END_REF]. Consequently, the metric spaces (E p ω , d p ) are uniformly convex for p > 1, giving them extra structure that will be explored in the latter parts of the paper: Theorem 1.1. Let p ∈ (1, ∞). Suppose that u ∈ E p ω , λ ∈ [0, 1] and [0, 1] ∋ t → v t ∈ E p ω is a finite energy geodesic segment. Then the following hold:

(i) d p (u, v λ ) 2 ≤ (1 -λ)d p (u, v 0 ) 2 + λd p (u, v 1 ) 2 -(p -1)λ(1 -λ)d p (v 0 , v 1 ) 2 , if 1 < p ≤ 2. (ii) d p (u, v λ ) p ≤ (1 -λ)d p (u, v 0 ) p + λd p (u, v 1 ) p -λ p 2 (1 -λ) p 2 d p (v 0 , v 1 ) p , if 2 ≤ p.
In the particular case p = 2 this result recovers the inequalities of Calabi-Chen [START_REF] Calabi | The space of Kähler metrics[END_REF], however our proof of Theorem 1.1 is very different from the argument in [START_REF] Calabi | The space of Kähler metrics[END_REF], as the differentiation of d p metrics is problematic for p ∕ = 2.

It was pointed out in the comments following [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Theorem 4.17] that d 1 -geodesic segments connecting the different points of (E 1 ω , d 1 ) are not unique. However, as a consequence of the above result it follows that uniqueness of d p -geodesic segments does hold in case p > 1:

Theorem 1.2. Let p ∈ (1, ∞), and suppose that [0, 1] ∋ t → v t ∈ E p ω is the L p finite energy geodesic connecting v 0 , v 1 ∈ E p ω . Then t → v t is the only d p -geodesic connecting v 0 , v 1 , i.e., (E p ω , d p ) is a uniquely geodesic metric space. The metric geometry of geodesic rays. Next we explore the metric geometry of R p u , the space of finite energy L p geodesic rays emanating from a fixed potential u ∈ E p ω . As a convention, given p ∈ [1, ∞), a finite energy geodesic ray [0, ∞) ∋ t → u t ∈ E p ω with u 0 = u will be simply denoted by {u t } t ∈ R p u . In accordance with the metric space literature, two d p -rays [0, ∞) ∋ t → u t , v [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Remark 1.6], who proved this for p = 1 under restrictive conditions on the slope of the K-energy along {u t } t . Thus, we can introduce a natural parallelism operator P uv : R p u → R p v for any u, v ∈ E p ω . Moreover it is possible to introduce natural metric structures on R p u and R p v making this map an isometry:

Theorem 1.3. Let p ∈ [1, ∞). For any u ∈ E p ω , (R p u , d c u,p ) is a complete metric space. For any v ∈ E p ω the parallelism operator P uv : (R p u , d c u,p ) → (R p v , d c v,p
) is an isometry.

In this result, the d c u,p metric is called the chordal L p metric between two rays, defined by the following expression:

d c u,p ({u t } t , {v t } t ) := lim t→∞ d p (u t , v t ) t , {u t } t ∈ R p u , {v t } t ∈ R p u . (3)
That this limit exists and is finite follows from [START_REF] Apostolov | Hamiltonian 2-forms in Kähler geometry III, extremal metrics and stability[END_REF]. Though not necessarily treated as a metric in other works, [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Corollary 5.6], [13, Formula 1.2] also consider the expression on the right hand side of (3), in the slightly restrictive case of unit speed geodesic rays, and non-Archimedean metrics respectively (see also [START_REF] Berman | Regularity of weak minimizers of the K-energy and applications to properness and K-stability[END_REF]Lemma 3.1]). Moreover, one would think that the metrics of the graded filtrations defined in [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF]Section 3] should be related to the above concept as well.

It was pointed out recently that L 1 Mabuchi geometry can be defined for big classes as well [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF]. Using this, it is possible to introduce the metric space of weak L 1 rays in the big context (see [START_REF] Darvas | The metric geometry of singularity types[END_REF] where we embed singularity types into the space of L 1 rays).

By the last part of the above theorem, there is no new information gained by considering different starting points for rays, hence it makes sense to restrict attention to the space (R p ω , d c p ), representing the space of rays emanating from 0 ∈ H ω . The above theorem points out that d c p thus defined gives a complete metric on the space of all L p rays emanating from a fixed starting point, that includes the constant ray. In our next main result we point out that the resulting metric spaces have rich geometry:

Theorem 1.4. (R p ω , d c p
) is a geodesic metric space for any p ∈ [1, ∞). Additionally, the radial K-energy is convex along d c p -geodesic segments.

The radial K-energy is defined for any {u t } t ∈ R p ω , and is given by the expression

K{u t } := lim t→∞ K(u t ) t ,
where [START_REF] Berman | Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties[END_REF][START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF]. The radial K-energy is d c p -lsc, possibly equal to ∞, and in the setting of unit speed geodesics, its definition agrees with the ¥ invariant of [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]. Also, there is clear parallel with the non-Archimedean K-energy (see [START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF] and references therein).

K : E p ω → (-∞, ∞] is the extended K-energy of Mabuchi from
This theorem represents the radial version of [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Theorem 2] and [8, Theorem 1.2] (building on [START_REF] Berman | Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics[END_REF]). In slight contrast with previous speculations in the literature (see for example [START_REF] Boucksom | Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs[END_REF] or [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Definition 1.8]) it seems more natural to consider the space of all d p -rays, not just the ones that have d p -unit speed. Allowing for a bigger class of rays makes possible the construction of d c p -geodesic segments running between any two points of R p ω , with good convexity properties. Moreover, the convexity of the radial K-energy on R p ω could potentially be used to set up the study of optimal degenerations as a convex optimization problem (see [START_REF] Dervan | The Kähler-Ricci flow and optimal degenerations[END_REF]).

The d c p -geodesic segments constructed in the proof of the above theorem are called d c p -chords, as they are reminiscent of the classical chords in the chordal geometry of the unit sphere of R n (at least when restricting to d p -unit speed rays). In case p > 1, due to uniform convexity (Theorem 1.1), we will construct the d c p -chords directly. In case p = 1, in the absence of uniform convexity, the construction of d c 1 -chords is done using an approximation procedure, via our next main theorem.

We have R p ω ⊂ R p ′ ω for any p ′ ≤ p. More importantly, by the proof of Theorem 1.4, d c p -chords are automatically d c p ′ -chords as well, giving further evidence that it is more advantageous to consider the space of all rays, not just the ones with d p -unit speed. This latter fact again represents the radial version of a well known phenomenon for the family of metric spaces (E p ω , d p ), p ≥ 1, according to which geodesics are "shared" when comparing different classes. Though the space of d p -unit speed rays seems to exhibit a metric structure reminiscent of the Tits geometry attached to CAT(0) spaces [START_REF] Bridson | Metric spaces of non positive curvature[END_REF], none of the above properties hold for these structures.

Next we turn to approximation. The collection of geodesic rays {u t } t ∈ R 1 ω with u t ∈ L ∞ , t ≥ 0 will be denoted by R ∞ ω , and will be referred to as the set of geodesic rays with bounded potentials. In addition to having bounded potentials, the rays of R ∞ ω are actually t-Lipschitz, and they solve the geodesic equation of L p Mabuchi geometry in the weak Bedford-Taylor sense, as opposed to the rays of R p ω , p ∈ [1, ∞), that are only limits of solutions to such equations (See Section 2.1).

By H 1, 1 ω we will denote the set of potentials in PSH(X, ω) whose Laplacian (or whose complex Hessian) is bounded. Analogously, the collection of geodesic rays

{u t } t ∈ R 1 ω with u t ∈ H 1,
1 ω , t ≥ 0 will be denoted by R 1, 1 ω , and will be referred to as the set of geodesic rays with C 1, 1 potentials. The space of rays with bounded Hessian, denoted by R 1,1 ω , is defined similarly.

The next result points out that R ∞ ω is d c p -dense in R p ω for any p ∈ [1, ∞). Also, we show that R 1, 1 ω dense among rays with finite radial K-energy. In both cases one can approximate with converging radial K-energy:

Theorem 1.5. Let {u t } t ∈ R p ω with p ∈ [1, ∞).
The following hold: (i) There exists a sequence

{u j t } t ∈ R ∞ ω such that u j t ↘ u t , t ≥ 0, d c p ({u j t } t , {u t } t ) → 0 and K{u j t } → K{u t }. (ii) If K{u t } < ∞, then there exists a sequence {v j t } t ∈ R 1, 1 ω such that v j t ↘ u t , t ≥ 0, d c p ({v j t } t , {u t } t ) → 0 and K{v j t } → K{u t }.
It remains to be seen if the condition K{u t } < ∞ can be omitted in (ii). This theorem can be seen as a radial analog of [START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF]Theorem 1.3], perhaps also making progress on the variational program designed to attack the uniform YTD conjecture (see step (4) in [13, p. 2], c.f. [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF]Conjecture 2.5]). Time will tell exactly how our results will fit into this program, but now it is enough to show that some C 1, 1 rays can be approximated by rays induced by test configurations (with converging K-energy) to prove the uniform YTD conjecture.

As a first step in obtaining Theorem 1.5(i), in Theorem 4.5 we show that one can approximate by bounded geodesic rays with possibly diverging radial K-energy. The argument uses [START_REF] Ross | Analytic test configurations and geodesic rays[END_REF], and this will suffice in case K{u t } = ∞, since K{•} is d c p -lsc. However to obtain (i) in case K{u t } is finite, a much more delicate construction will be needed, building on the relative Ko󰀀 lodziej type estimate of [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]. The proof of (ii) builds on (i), and novel apriori C 1, 1 estimates along geodesic segments that are "scalable" along rays. These will be obtained using the framework of [START_REF] He | On the space of Kähler potentials[END_REF] and [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF].

Applications to geodesic stability. We point out applications to existence of constant scalar curvature Kähler (csck) metrics in terms of geodesic stability, going back to Donaldson's early conjectures in [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF].

To start, we say that (X, ω) is geodesically

L p /C 1, 1-semistable if for any {u t } t ∈ R p ω /R 1, 1 ω we have that K{u t } ≥ 0 for p ∈ [1, ∞].
Regarding the relevance of semistability for the csck continuity method, we refer to [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]. As an immediate consequence of Theorem 1.5 we obtain the following:

Theorem 1.6. (X, ω) is geodesically L 1 -semistable if and only if it is geodesically C 1, 1- semistable.
Let G := Aut 0 (X) be the identity component of the group of holomorphic automorphisms of X. By I : E 1 ω → R we denote the Monge-Ampère energy functional (sometimes called Aubin-Yau or Aubin-Mabuchi energy). Then, as explained in [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF], G induces an isometry on E 1 0 = E 1 ω ∩ I -1 (0), and one can introduce the following pseudo-metric on the orbits E 1 0 /G: d 1,G (Gu 0 , Gu 1 ) := inf g∈G d 1 (u 0 , g.u 1 ).

Moreover, one can analogously define the space of normalized rays

R p /R 1, 1/R 1,1 , p ∈ [1, ∞],
where we restrict to rays

{u t } t ∈ R p ω /R 1, 1 ω /R 1,1
ω with I(u t ) = 0, t ≥ 0. By showing that minimizers of the K-energy on E 1 ω are actually smooth csck potentials [26, Theorem 1.5], Chen-Cheng have verified the last remaining condition of the existence/properness principle of [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF], applied to the case of csck metrics. Together with the necessity result [9, Theorem 1.5], their theorem showed that existence of csck metrics is equivalent with properness of K in the following sense: there exists δ, γ > 0 such that

K(u) ≥ δd 1,G (G0, Gu) -γ, u ∈ E 1 ω , (4)
Clearly, d 1,G (Gv 0 , Gv 1 ) ≤ d 1 (v 0 , v 1 ), v 0 , v 1 ∈ E 1 ω , and we say that {u t } t ∈ R 1 is Gcalibrated if the curve t → Gu t is a d 1,G -geodesic with the same speed as {u t } t , i.e.,

d 1,G (Gu 0 , Gu t ) = d 1 (u 0 , u t ), t ≥ 0. Geometrically, {u t } t is G-calibrated if it cuts each G-orbit inside E 1
ω "perpendicularly". In case G = {Id}, every ray is G-calibrated.

Building on these concepts, it is natural to state the C 1,1 uniform analog to Donaldson's geodesic stability conjecture, with the original formulation of [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]Conjecture 12] more closely related to the language of "polystability": Conjecture 1.7 (C 1,1 uniform geodesic stability). Let (X, ω) be a compact Kähler manifold. Then the following are equivalent: (i) There exists a csck metric in H.

(ii) There exists δ > 0 such that K{u t } ≥ δ lim sup t d 1,G (G0,Gut) t for all geodesic rays {u t } t ∈ R 1,1 . (iii) K is G-invariant and there exists δ > 0 such that for all G-calibrated geodesic rays {u t } t ∈ R 1,1 we have that K{u t } ≥ δd 1 (0, u 1 ).

The statement of (ii) clearly points out that uniform geodesic stability is simply the condition that tests energy properness (expressed in (4)) along a class of geodesic rays.

As the notion of G-calibrated rays has an obvious analog in case of the space of finite dimensional rays as well (within the context of Kähler quantizaton), we included this condition here to perhaps facilitate in the future an alternative definition for uniform K-stability in the presence of vector fields.

As explained in [44, Proposition 5.5], in the above conjecture the d 1 distance is interchangeable with Aubin's J functional. Lastly, given that rays induced by 1-parameter actions of G are never G-calibrated, the condition that K is G-invariant (equivalent to vanishing Futaki invariant [START_REF] Futaki | An obstruction to the existence of Einstein Kähler metrics[END_REF]) is necessary in the statement of (iii).

Using our above theorems, we prove in Theorem 6.2 and Theorem 6.3 that the C Clearly, given the obvious inclusions among classes or geodesic rays, the L p versions of Conjecture 1.7 follow as well (with R p replacing R 1, 1 in the statement). Though slightly different in formulation, the L ∞ version of this result essentially confirms the equivalences between the conditions (3), ( 4) and ( 5) in [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Question 1.12] (see also the closely related questions of [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF]Remark 1.3]). In case G = {Id}, the statement of the theorem can be made especially simple: Theorem 1.9. Let (X, ω) be a compact Kähler manifold without non-trivial holomorphic vector fields. Then the following are equivalent: (i) There exists a csck metric in H. (ii) There exists δ > 0 such that K{u t } ≥ δd 1 (0, u 1 ) for all {u t } t ∈ R 1, 1.

It remains to be seen if in the above stability results one can use rays that have potentials with fully bounded Hessian, not just bounded complex Hessian. Even if possible, this small improvement seems to require substantial amount of new work. Further optimizations are extremely unlikely, given that the typical regularity of geodesics breaks down beyond C 2 estimates. One would think that generalizations to the context of extremal and conical type csck metrics should be possible, using our results together with [START_REF] He | On Calabi's extremal metric and properness[END_REF][START_REF] Zheng | Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability[END_REF].

Connections with the literature. Uniform convexity of metric spaces is an active area of research (see [START_REF] Ohta | Convexities of metric spaces[END_REF][START_REF] Kell | Uniformly convex metric spaces[END_REF][START_REF] Kuwae | Jensen's inequality on convex spaces[END_REF][START_REF] Naor | Poincaré inequalities, embeddings, and wild groups[END_REF] and references therein). In particular, by [65, Proposition 2.5] the inequalities of Theorem 1.1 are essentially optimal.

The notion of K-stability goes back to work of Tian [START_REF] Tian | Kähler-Einstein metrics with positive scalar curvature[END_REF], with generalizations and precisions made along the way by S. Donaldson [START_REF] Donaldson | Scalar curvature and stability of toric varities[END_REF], Li-Xu [START_REF] Li | Special test configurations and K-stability of Fano varieties[END_REF], G. Székelyhidi [START_REF] Székelyhidi | Extremal metrics and K-stability[END_REF] and many others. Though the precise form of K-stability is still not fully clarified for general Kähler manifolds [START_REF] Apostolov | Hamiltonian 2-forms in Kähler geometry III, extremal metrics and stability[END_REF], at least in the absence of non-trivial holomorphic vector fields, it is widely expected that uniform K-stability will be equivalent with existence of csck metrics (see [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]Question 1.12], [13, Conjecture 4.9]). Informally, uniform K-stability simply says that Conjecture 1.7 holds for C 1, 1 rays that are induced by the so called test configurations of (X, ω).

Closing the gap between L 1 uniform geodesic stability and uniform K-stability is the last remaining step in the variational program designed to attack the uniform YTD conjecture (see [13, p.2]), with our Theorem 1.8 representing an intermediate step. To facilitate further progress in this direction, based on the findings of Theorem 1.4, one possible approach would be to develop the radial analog of the Kähler quantization scheme, recently extended to the d p -metric completions in [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF] (building on prior work by Berndtsson [START_REF] Berndtsson | Probability measures related to geodesics in the space of Kähler metrics[END_REF], Chen-Sun [START_REF] Chen | Space of Kähler metrics (V)-Kähler quantization[END_REF], Donaldson [START_REF] Donaldson | Scalar curvature and stability of toric varities[END_REF][START_REF] Donaldson | Scalar curvature and projective embeddings, II[END_REF], Phong-Sturm [START_REF] Phong | The Monge-Ampère operator and geodesics in the space of Kähler potentials[END_REF], Song-Zeldtich [START_REF] Song | Bergman metrics and geodesics in the space of Kähler metrics on toric varieties[END_REF], Tian [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF] and others). Indeed, in case the Kähler structure (X, ω) is induced by an ample Hermitian line bundle (L, h), it is pointed out in [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF][START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF][START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF] that R k ω , the space of finite dimensional geodesic rays associated to the space of Hermitian metrics H k ω on H 0 (X, L k ) admits a natural metric d c p,k , likely representing the finite dimensional analog of our d c p metrics. If one could show in the spirit of [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF]Theorem 1.1] that the metric spaces (R k ω , d c p,k ) approximate (R p ω , d c p ) (or relevant parts of it) in the large k-limit, then that would open the door for a version of Theorem 1.5, where the rays from R 1, 1 ω are replaced by rays induced by test configurations. Even if successful, it is not clear how convergence of the radial K-energy can be achieved (see [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF]Conjecture 2.5]), and for the difficulties that need to be overcome in this approach we refer to the comments following [START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF]Conjecture 4.9].

Further connections with geodesic rays are explored in [START_REF] Darvas | The metric geometry of singularity types[END_REF], related to the metric geometry of the space of singularity types, and complex Monge-Ampère equations with prescribed singularity.

Organization of the paper. In Section 2 we recall basic facts about the L p Mabuchi geometry of the space of Kähler metrics, the relative Kolodziej type estimate of [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF], and we prove weighted versions of the classical inequalities of Clarkson and Ball-Carlen-Lieb that will be needed later. In Section 3 we prove Theorems 1.1 and 1.2 regarding uniform convexity, and uniqueness of geodesics in L p Mabuchi geometry when p > 1. In Section 4 we study the chordal L p metric structures on the space of geodesic rays and prove Theorem 1.4. In Section 5 we prove Theorem 1.5, our main approximation result, and in Section 6 we show that the C 1, 1 version of the uniform geodesic stability conjecture holds.

Acknowledgements. We would like to thank Martin Kell and Alexander Lytchak for enlightening discussions on various notions of convexity in metric spaces. We also thank László Lempert for his suggestions that improved the presentation of the paper. We thank the referee for numerous useful remarks and suggestions which improve the presentation of the paper. This work was partially supported by NSF grant 1610202.

Preliminaries

The L p Finsler geometry of the space of Kähler potentials

In this short section we recall the basics of finite energy pluripotential theory, as introduced by Guedj-Zeriahi [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF], and the Finsler geometry of the space of Kähler potentials, as introduced by the first author [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]. For a detailed account on these matters we refer to the recent textbook [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF] and lecture notes [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF].

As a matter of convention for the duration of the paper we denote by V the total volume of the Kähler class [ω]:

V := 󰁝 X ω n .
By PSH(X, ω) we denote the space of ω-plurisubharmonic (ω-psh) functions. Extending the ideas of Bedford-Taylor, Guedj-Zeriahi introduced the non-pluripolar Monge-Ampère mass for a general potential u ∈ PSH(X, ω) as the following limit [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF]:

ω n u := lim k→∞ {u>-k} (ω + √ -1∂ ∂ max(u, -k)) n .
For such measures one has an estimate on the total mass

󰁕 X ω n u ≤ 󰁕 X ω n = V
, and E ω is the set of potentials with full/maximum mass:

E ω := {u ∈ PSH(X, ω) s.t. 󰁕 X ω n u = 󰁕 X ω n = V }.
Furthermore, potentials u ∈ E ω that satisfy an L p type integral condition are members of the so called finite-energy spaces of [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF]:

E p ω = 󰁱 u ∈ E ω s.t. 󰁝 X |u| p ω n u < +∞ 󰁲 .
Now we recall some of the main points on the L p Finsler geometry of the space of Kähler potentials. By definition, the space of Kähler potentials H ω is an open convex subset of C ∞ (X), hence one can think of it as a trivial Fréchet manifold. As a result, one can introduce on H ω a collection of L p type Finsler metrics. If u ∈ H ω and ξ ∈ T u H ω ≃ C ∞ (X), then the L p norm of ξ is given by the following expression:

󰀂ξ󰀂 p,u = 󰀕 1 V 󰁝 X |ξ| p ω n u 󰀖 1 p .
In case p = 2, this construction reduces to the Riemannian geometry of Mabuchi [START_REF] Mabuchi | Some symplectic geometry on compact Kähler manifolds I[END_REF] (independently discovered by Semmes [START_REF] Semmes | Complex Monge-Ampère and symplectic manifolds[END_REF] and Donaldson [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]). Using these Finsler metrics, one can introduce path length metric structures (H ω , d p ). In [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Theorem 2], the first author identified the completion of these spaces with E p ω ⊂ PSH(X, ω) from above, and it turns out that (E p ω , d p ) is a complete geodesic metric space. The distinguished d p -geodesic segments of the completion (E p ω , d p ) are constructed as upper envelopes of quasi-psh functions, as we now elaborate. Let S = {0 < Re s < 1} ⊂ C be the unit strip, and π S×X : S × X → X denotes projection to the second component.

We consider u 0 , u 1 ∈ E p ω . We say that the curve [0

, 1] ∋ t → v t ∈ E p ω is a weak subgeodesic connecting u 0 , u 1 if d p (v t , u 0,1 ) → 0 as t → 0, 1, and the extension v(s, x) = v Re s (x) is π * ω-psh on S × X, i.e., π * ω + i∂ S×X ∂S×X v ≥ 0, as currents on S × X.
As shown in [START_REF] Darvas | The Mabuchi completion of the space of Kähler potentials[END_REF][START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]

, a distinguished d p -geodesic [0, 1] ∋ t → u t ∈ E p
ω connecting u 0 , u 1 can be obtained as the supremum of all weak subgeodesics:

u t := sup{v t | t → v t is a subgeodesic connecting u 0 , u 1 }, t ∈ [0, 1]. ( 5
)
Given u 0 , u 1 ∈ E p ω , we call (5) the L p finite energy geodesic (or simply finite energy geodesic) connecting u 0 , u 1 .

In case the endpoints u 0 , u 1 are from H ω , the finite energy geodesic connecting them is actually C 1,1 on S × X, as shown by Chu-Tosatti-Weinkove [START_REF] Chu | On the C 1,1 regularity of geodesics in the space of Kähler metrics[END_REF]. In this paper we only need the C 1, 1 regularity of geodesics connecting smooth endpoints which had been previously proved by Chen [START_REF] Chen | The space of Kähler metrics[END_REF] (for a survey see B󰀀 locki [START_REF] Blocki | The complex Monge-Ampère equation in Kähler geometry[END_REF]). His proof relies on the so called ε-geodesic that we now recall (for a survey see [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Section 3

.1]). Given u 0 , u 1 ∈ H ω , by u ε ∈ C ∞ ([0, 1] × X) we denote the smooth ε-geodesic connecting u 0 , u 1 , i.e., [0, 1] ∋ t → u ε t ∈ H ω solving the following elliptic PDE on [0, 1] × X: 󰀃 üε t -|∇ ut | 2 ω u ε t 󰀄 ω n u ε t ω n = ε, u ε 0 := u 0 , u ε 1 := u 1 . (6)
Note that t 󰀁 → u ε t is a subgeodesic connecting u 0 , u 1 . Given that the complex Hessian of u ε is bounded on [0, 1] × X [START_REF] Chen | The space of Kähler metrics[END_REF], one can take the limit ε → 0, to obtain

u ∈ C 1, 1([0, 1] × X), the C 1, 1-geodesic connecting u 0 , u 1 : [0, 1] ∋ t → u t ∈ H 1, 1 ω . (7)
As shown in [59, Theorem

1.1], if one merely has u 0 , u 1 ∈ H 1, 1 ω , the curve in (7) still exists u t ∈ H 1, 1 ω for all t ∈ [0, 1], however it is not known if the total Laplacian of u on [0, 1] × X is bounded.
Due to the "Perron type" definition of geodesics ( 5), finite energy geodesic segments satisfy a comparison principle. In particular, we note the following simple consequence of the comparison principle for geodesics.

Lemma 2.1. Let u ε be the smooth ε-geodesic connecting u 0 , u 1 ∈ H ω and u t be the C 1, 1 geodesic connecting u 0 , u 1 . Then u ε ≤ u.

Regarding the metric d p the following double estimate holds for some dimensional constant C > 1 and all p ≥ 1 [33, Theorem 3]:

1 C d p (u 0 , u 1 ) p ≤ 1 V 󰁝 X |u 0 -u 1 | p ω n u 0 + 1 V 󰁝 X |u 0 -u 1 | p ω n u 1 ≤ Cd p (u 0 , u 1 ) p , u 0 , u 1 ∈ E p ω .
We recall that for any u ∈ PSH(X, ω) there exists u j ∈ H ω such that u j decreases to u. This is a result due to Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF] with a simpler proof due to B󰀀 locki-Ko󰀀 lodziej [START_REF] B󰀀 Locki | On regularization of plurisubharmonic functions on manifolds[END_REF]. It is well known that the Monge-Ampère energy I : E 1 ω → R defined by

I(u) = 1 V (n + 1) n 󰁛 j=0 󰁝 X uω n-j ∧ ω j u
is affine along finite energy geodesics [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]. Moreover, the same is true for sup X u t in case u 0 = 0:

Lemma 2.2. Let [0, 1] ∋ t → u t ∈ E 1
ω be a finite energy geodesic with u 0 = 0. Then t → sup X u t is affine.

As E p ω ⊂ E 1 ω for all p ≥ 1, Lemma 2.2 is true in E p ω , p ≥ 1,
as well. This is essentially [32, Theorem 1](ii), that is stated for bounded geodesics. Since finite energy geodesic segments can be approximated decreasingly by bounded geodesic segments, the above result follows as a consequence of Hartogs' lemma [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF]Proposition 8.4]. For more on L p Mabuchi geometry we refer to [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Chapter 3].

The K-energy and constant scalar curvature metrics

Given a smooth closed (1, 1)-form χ the χ-contracted version of the Monge-Ampère energy is defined as

I χ (u) = 1 nV n-1 󰁛 j=0 󰁝 X uω j u ∧ ω n-1-j ∧ χ, u ∈ E 1 ω .
It was shown in [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Lemma 4.15] and [8, Section 4] that the energy functionals I χ can be extended to (E 1 ω , d 1 ) as d 1 -Lipschitz functionals. We recall the definition of the I-functional introduced by Aubin [START_REF] Aubin | Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration dúne inégalité (French)[END_REF] (and extended to E 1 ω in [START_REF] Berman | Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties[END_REF]):

I(u 0 , u 1 ) = 1 V 󰁝 X (u 0 -u 1 )(ω n u 1 -ω n u 0 ), u 0 , u 1 ∈ E 1 ω .
From the definition it is clear that I is symmetric and invariant under adding constants. By an integration by parts we see that I is non-negative.

The (extended) K-energy K : E 1 → (-∞, ∞] is defined as follows:

K(u) = Ent(ω n , ω n u ) + SI(u) -nI Ric(ω) (u), u ∈ E 1 ω . (8)
where Ent(ω n , ω n u ) is the entropy of the measure ω n u with respect to ω n :

Ent(ω n , ω n u ) = V -1 󰁝 X log(ω n u /ω n )ω n u ,
and S = 1 V 󰁕 X S ω ω n is the average scalar curvature, that can be seen to be independent of the choice of background metric. When restricted to H ω , the above formula for the K-energy was originally introduced by Chen-Tian [START_REF] Chen | On the lower bound of the Mabuchi energy and its application[END_REF], with a similar formula already appearing in [START_REF] Tian | The K-energy on hypersurfaces and stability[END_REF].

The first order variation of K is given by the following formula:

〈DK(u), δv〉 = V -1 󰁝 X δv( S -S ωu )ω n u ,
where S = nV -1 󰁕 X Ric(ω) ∧ ω n-1 . Hence, the critical points of K are the constant scalar curvature potentials, as these satisfy S -S ωu = 0. It was proved in [START_REF] Berman | Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics[END_REF] that the K-energy K is convex along C 1, 1 geodesics. As a consequence cscK potentials are minimizers of K.

The relative Ko󰀀 lodziej type estimate

In this short subsection we recall the basics of relative pluripotential theory that are needed to state the relative Ko󰀀 lodziej type estimates of [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]. For more details we refer to the sequence of papers [START_REF] Darvas | On the singularity type of full mass currents in big cohomology classes[END_REF][START_REF] Darvas | Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity[END_REF][START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF][START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF].

Let E be a Borel subset of X. Given χ ∈ PSH(X, ω), we define the χ-relative capacity of E as

Cap χ (E) := sup 󰀝󰁝 E ω n u ; u ∈ PSH(X, ω), χ -1 ≤ u ≤ χ 󰀞 . (9)
When χ = 0, we recover the classical Monge-Ampère capacity Cap ω (see e.g. [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]). For more on this concept we refer to [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]Section 4].

Given u ∈ PSH(X, ω), we recall the definition of envelopes with respect to singularity type, introduced by Ross and Witt Nyström [START_REF] Ross | Analytic test configurations and geodesic rays[END_REF]:

P [u] := usc 󰀓 lim C→+∞ P (0, u + C) 󰀔 ∈ PSH(X, ω),
where P (φ, ψ) := sup{v ∈ PSH(X, ω) s.t. v ≤ φ and v ≤ ψ}. In addition to appearing in the statement of the relative Ko󰀀 lodziej type estimate below, this concept also plays a role in Theorem 4.5, where it is used to approximate geodesic rays, via [START_REF] Ross | Analytic test configurations and geodesic rays[END_REF].

Finally we recall the following L ∞ estimate from [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]:

Theorem 2.3. [39, Theorem 3.3] Let a ∈ [0, 1), A > 0, χ ∈ PSH(X, θ) and 0 ≤ f ∈ L p (X, ω n ) for some p > 1. Assume that u ∈ PSH(X, θ), normalized by sup X u = 0, satisfies θ n u ≤ f ω n + aθ n χ . (10)
Assume also that

󰁝 E f ω n ≤ A[Cap χ (E)] 2 , ( 11
)
for every Borel subset E ⊂ X. If P [u] is less singular than χ then χ -sup X χ -C 󰀓 󰀂f 󰀂 L p , p, (1 -a) -1 , A 󰀔 ≤ u.
Here, given two potentials u, v ∈ PSH(X, ω), we say that u is less singular than v if u ≥ v -C, for some constant C.

This theorem generalizes the classical estimates of Ko󰀀 lodziej from [START_REF] Ko󰀀 | The complex Monge-Ampère equation[END_REF], and it is used in [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF] to solve complex Monge-Ampère equations with prescribed singularity type, and to resolve the log-concavity conjecture of the volume in pluripotential theory. Here we will use it in Section 5 to show that it is possible to approximate L p geodesic rays with bounded ones that have converging radial K-energy.

Weighted Clarkson and Ball-Carlen-Lieb type inequalities

In this short preliminary section we point out relevant extensions of well known inequalities due to Clarkson [START_REF] Clarkson | Uniformly convex spaces[END_REF] and Ball-Carlen-Lieb [START_REF] Ball | Sharp uniform convexity and smoothness inequalities for trace norms[END_REF] for L p spaces, introducing a weight λ ∈ [0, 1] into these results. These theorems are almost certainly well known to experts in analysis, but we could not find the versions below in the literature.

Theorem 2.4. Suppose that p ≥ 2, λ ∈ [0, 1] and f, g ∈ L p (ν), where ν is a measure on the set X. Then

λ󰀂f 󰀂 p p + (1 -λ)󰀂g󰀂 p p ≥ 󰀂λf + (1 -λ)g󰀂 p p + λ p 2 (1 -λ) p 2 󰀂f -g󰀂 p p . (12)
Proof. Since t → |t| p 2 is a convex function, we can write the following estimates:

λ󰀂f 󰀂 p p + (1 -λ)󰀂g󰀂 p p ≥ 󰁝 X (λf 2 + (1 -λ)g 2 ) p 2 dν = 󰁝 X ((λf + (1 -λ)g) 2 + λ(1 -λ)(f -g) 2 ) p 2 dν ≥ 󰁝 X (|λf + (1 -λ)g| p + λ p 2 (1 -λ) p 2 |f -g| p )dν,
where in the last step we have used that (a 2 + b 2 )

1 2 ≥ (a p + b p ) 1 p , a, b ≥ 0. Theorem 2.5. Suppose that 1 < p ≤ 2, λ ∈ [0, 1] and f, g ∈ L p (ν)
, where ν is a measure on the set X. Then

λ󰀂f 󰀂 2 p + (1 -λ)󰀂g󰀂 2 p ≥ 󰀂λf + (1 -λ)g󰀂 2 p + (p -1)λ(1 -λ)󰀂f -g󰀂 2 p . ( 13
)
Proof. The proof will be given using diadic approximation. Indeed, it is enough to prove [START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF]

for λ = k 2 m , k, m ∈ N with 1 ≤ k ≤ 2 m .
We will argue by induction on m. For m = 1 and k = 0, 1, 2, the statement of ( 13) is either a triviality or reduces to [START_REF] Ball | Sharp uniform convexity and smoothness inequalities for trace norms[END_REF]Proposition 3]. Let us assume that m > 1 and the statement holds for m -1. We can assume that k is odd, as otherwise the inequality reduces to the case m -1. Using [3, Proposition 3], we start with the following estimate:

1 2 󰀐 󰀐 󰀐 k -1 2 m f + 󰀓 1 - k -1 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p + 1 2 󰀐 󰀐 󰀐 k + 1 2 m f + 󰀓 1 - k + 1 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p ≥ (14)
≥ 󰀐 󰀐 󰀐 k 2 m f + 󰀓 1 - k 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p + (p -1) 󰀐 󰀐 󰀐 f 2 m - g 2 m 󰀐 󰀐 󰀐 2 p .
Since both k + 1 and k -1 are even, by the inductive step we also have that:

k + 1 2 m 󰀂f 󰀂 2 p + 󰀓 1 - k + 1 2 m 󰀔 󰀂g󰀂 2 p ≥ (15)
≥ 󰀐 󰀐 󰀐 k + 1 2 m f + 󰀓 1 - k + 1 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p + (p -1) k + 1 2 m 󰀓 1 - k + 1 2 m 󰀔 󰀂f -g󰀂 2 p , k -1 2 m 󰀂f 󰀂 2 p + 󰀓 1 - k -1 2 m 󰀔 󰀂g󰀂 2 p ≥ (16)
≥ 󰀐 󰀐 󰀐 k -1 2 m f + 󰀓 1 - k -1 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p + (p -1) k -1 2 m 󰀓 1 - k -1 2 m 󰀔 󰀂f -g󰀂 2 p .
Adding ( 15) and ( 16) and then using ( 14) we arrive at

k 2 m 󰀂f 󰀂 2 p + 󰀓 1 - k 2 m 󰀔 󰀂g󰀂 2 p ≥ 󰀐 󰀐 󰀐 k 2 m f + 󰀓 1 - k 2 m 󰀔 g 󰀐 󰀐 󰀐 2 p + (p -1) k 2 m 󰀓 1 - k 2 m 󰀔 󰀂f -g󰀂 2 p ,
what we desired to prove.

Remark 2.6. As alluded to at the beginning of the subsection, in case λ = 1 2 , Theorem 2.4 and Theorem 2.5 recover the well known inequalities of Clarkson [START_REF] Clarkson | Uniformly convex spaces[END_REF] and Ball-Carlen-Lieb [3, Proposition 3] respectively.

Uniform convexity and uniqueness of geodesics

Before proving the main result of this section, we first point out the following result about the "spread" of geodesic segments in E p ω , sharing a common smooth endpoint:

Theorem 3.1. Suppose that p ≥ 1, u ∈ H ω and [0, l] ∋ t → u t , v t ∈ E p ω are two finite energy geodesic segments with u = u 0 = v 0 , and l ∈ R + . Then 󰀗 󰁝 X | u0 -v0 | p ω n u 󰀘 1 p ≤ d p (u t , v t ) t , t ∈ [0, l]. (17)
Proof. We first assume that u l ≥ v l . Then we note that (thanks to convexity) u t ≥ v t for all t ∈ [0, l]. Furthermore, using d p -approximation of the endpoints u l , v l ∈ E p ω by decreasing sequences of potentials in H ω , it is enough to prove [START_REF] Boucksom | Uniform K-stability and asymptotics of energy functionals in Kähler geometry[END_REF] for

C 1, 1-geodesics t → u t , v t with u l , v l ∈ H ω (see [8, Proposition 4.3]).
Using the convexity condition (1) and [33, Lemma 5.1] for 0 ≤ s ≤ t ≤ l we have

d p (u t , v t ) p t p ≥ d p (u s , v s ) p s p ≥ 󰁝 X (u s -v s ) p s p ω n us .
As s → 0 + , using the fact that the geodesics are C 1, 1, we get that (u sv s ) p /s p uniformly converges to ( u0 -v0 ) p which is a continuous function on X. Since ω n us → ω n u weakly (see [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Theorem 5

(i)]) it follows that d p (u t , v t ) p t p ≥ 󰁝 X ( u0 -v0 ) p ω n u .
We now treat the general case, when u l and v l may not be comparable. By the previous step, for t ∈ [0, l] we have

d p (u t , P (u t , v t )) p t p ≥ 󰁝 X | u0 -ẇt 0 | p ω n u and d p (v t , P (u t , v t )) p t p ≥ 󰁝 X | v0 -ẇt 0 | p ω n u ,
where [0, t] ∋ s 󰀁 → w t s ∈ E p ω is the finite energy geodesic connecting w 0 := u 0 and w t := P (u t , v t ).

Due to the comparison principle for geodesics, we note that ẇt 0 ≤ u0 , v0 . Using the Pythagorean formula [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Corollary 4.14] and the inequality a p + b p ≥ max(a p , b p ) ≥ |a -b| p , a, b ≥ 0, we can sum up the above inequalities to arrive at the conclusion:

d p (u t , v t ) p t p = d p (u t , P (u t , v t )) p t p + d p (v t , P (u t , v t )) p t p ≥ 󰁝 X | u0 -v0 | p ω n u , t ∈ [0, l].
Before proceeding we note that Theorem 3.1 implies the following Lidskii type inequality proved in the case of Hodge type Kähler metrics in [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF]:

Corollary 3.2. If α, β, γ ∈ E p ω , p ≥ 1, with α ≥ β ≥ γ then: d p (β, γ) p ≤ d p (α, γ) p -d p (α, β) p .
Proof. By density it is enough to show this estimate for α, β, γ ∈ H ω . Let [0, 1] ∋ t → u t , v t ∈ E p ω be the increasing/decreasing C 1, 1-geodesics joining u 0 := β, u 1 := α and v 0 := β, v 1 := γ respectively. Then, due to t-monotonicity, v0 ≤ 0 ≤ u0 . Then by Theorem 3.1 and [33, Theorem 1], the following holds:

d p (α, γ) p = d p (u 1 , v 1 ) p ≥ 󰁝 X | u0 -v0 | p ω n β ≥ 󰁝 X (| u0 | p + | v0 | p ) ω n β = d p (α, β) p + d p (γ, β) p .
Next we prove the main result of this section about the uniform convexity of the spaces (E p ω , d p ) for p > 1. This will follow after an adequate combination of Theorem 3.1 and the extension of the inequalities of Clarkson and Ball-Carlen-Lieb, obtained in the previous section.

Theorem 3.3. Suppose that u ∈ E p ω , λ ∈ [0, 1] and [0, 1] ∋ t → v t ∈ E p
ω is a finite energy geodesic segment. Then the following hold:

(i) d p (u, v λ ) 2 ≤ (1 -λ)d p (u, v 0 ) 2 + λd p (u, v 1 ) 2 -(p -1)λ(1 -λ)d p (v 0 , v 1 ) 2 , if 1 < p ≤ 2. (ii) d p (u, v λ ) p ≤ (1 -λ)d p (u, v 0 ) p + λd p (u, v 1 ) p -λ p 2 (1 -λ) p 2 d p (v 0 , v 1 ) p , if 2 ≤ p.
that u, v 0 , v 1 ∈ H ω and hence t → v t is C 1, 1. Fixing ε > 0 momentarily, let [0, 1] ∋ t → v ε t ∈ H ω be Chen's smooth ε-geodesic connecting v 0 , v 1 ∈ H ω , see Section 2.1. Moreover, let [0, 1] ∋ t → α λ,ε t ∈ E p ω be the C 1, 1 geodesic connecting u and v ε λ . Let [0, λ] ∋ t → h ε t ∈ E p ω be the C 1, 1 geodesic connecting v 0 and v ε λ . Similarly, let [λ, 1] ∋ t → k ε t ∈ E p ω be the C 1, 1 geodesic connecting v ε λ and v 1 .
We now assume that 2 ≤ p to address (ii). Using Theorem 3.1 twice, for pairs of geodesics emanating from v ε λ , we conclude that 󰁝

X 󰀏 󰀏 αλ,ε 1 -λ ḣε λ 󰀏 󰀏 p ω n v ε λ ≤ d p (u, v 0 ) p , 󰁝 X 󰀏 󰀏 αλ,ε 1 + (1 -λ) kε λ 󰀏 󰀏 p ω n v ε λ ≤ d p (u, v 1 ) p .
In the first inequality, we have applied Theorem 3.1 for u 0 = v 0 = v ε λ (hence the geodesic segments h ε t and α λ,ε t are both parametrized in the opposite directions). In the second inequality we have applied Theorem 3.1 for u 0 = v 0 = v ε λ (hence the geodesic segment k ε t is parametrized in the positive direction while α λ,ε t is parametrized in the opposite direction, that's why we have plus signs).

By the comparison principle for geodesics, we have that

v ε t ≤ h ε t ≤ v t , t ∈ [0, λ] and v ε t ≤ k ε t ≤ v t , t ∈ [λ, 1] (see Lemma 2.1)
. Again, by the comparison principle, the concatenation of t → h ε t and t → k ε t is t-convex and we obtain that ḣε λ → vλ and kε λ → vλ uniformly on X. Using this and the above two estimates we can write:

(1 -λ)d p (u,v 0 ) p + λd p (u, v 1 ) p ≥ 󰁝 X (1 -λ)| αλ,ε 1 -λ ḣε λ | p + λ| αλ,ε 1 + (1 -λ) kε λ | p ω n v ε λ ≥ (1 -λ) 󰁝 X | αλ,ε 1 -λ vλ | p ω n v ε λ + λ 󰁝 X | αλ,ε 1 + (1 -λ) vλ | p ω n v ε λ -O(ε) ≥ 󰁝 X | αλ,ε 1 | p ω n v ε λ + λ p 2 (1 -λ) p 2 󰁝 X | vλ | p ω n v ε λ -O(ε) = d p (u, v ε λ ) p + λ p 2 (1 -λ) p 2 󰁝 X | vλ | p ω n v ε λ -O(ε), (18)
where in the third line we have used Theorem 2.4, and in the last line we have used [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Theorem 1]. Letting ε → 0, since ω n v ε λ ⇀ ω n v λ and O(ε) → 0, another application of [33, Theorem 1] gives (ii). Now we assume that 1 < p ≤ 2 and we address the inequality of (i). The proof is exactly the same, except for [START_REF] Boucksom | Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs[END_REF], where we use the estimate of Theorem 2.5 instead of Theorem 2.4.

Remark 3.4. Suppose that ω is the curvature of a Hermitian line bundle (L, h). By exactly the same arguments, one can show that the inequalities of Theorem 3.3 also hold for the finite dimensional L p type metric spaces (H k ω , d p,k ), as considered in [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF]. Using the quantization scheme of this paper [43, Theorem 1.2], an alternative proof of Theorem 3.3 can be thus given when [ω] is integral.

Finally we point out that using the above result one can show that the finite energy geodesic segments of E p ω are the only metric geodesics when p > 1:

Theorem 3.5. Let p ∈ (1, ∞), and suppose that [0, 1] ∋ t → v t ∈ E p ω is the finite energy geodesic connecting v 0 , v 1 ∈ E p ω . Then t → v t is the only d p -geodesic connecting v 0 , v 1 . Proof. Suppose that [0, 1] ∋ t → u t ∈ E p ω is a d p -geodesic connecting v 0 , v 1 ,
∕ = v t , Theorem 3.3 implies that d p (v 0 , h t ) < max{d p (v 0 , u t ), d p (v 0 , v t )} = td p (v 0 , v 1 ). Similarly, d p (v 1 , h t ) < max{d p (v 1 , u t ), d p (v 1 , v t )} = (1 -t)d p (v 0 , v 1 ). The triangle inequality now gives a contradiction, implying that u t = v t , t ∈ [0, 1].
A more careful analysis of the above proof yields the following:

Proposition 3.6. Suppose that p > 1 and [0, 1] ∋ l → u l ∈ E p ω is a finite energy geodesic. Let v ∈ E p ω such that d p (v, u 0 ) ≤ (t + ε)d p (u 0 , u 1 ) and d p (v, u 1 ) ≤ (1 -t + ε)d p (u 0 , u 1 )
for some ε > 0 and t ∈ [0, 1]. Then there exists C(p) > 0 such that

d p (v, u t ) ≤ ε 1 r Cd p (u 0 , u 1 ),
where r := max(2, p).

Proof. Let h be the d p -midpoint of the finite energy geodesic connecting v and u t . Then Theorem 3.3 implies that

d p (u 0 , h) ≤ 󰁫 1 2 d p (u 0 , v) r + 1 2 d p (u 0 , u t ) r -cd p (v, u t ) r 󰁬 1 r , d p (u 1 , h) ≤ 󰁫 1 2 d p (u 1 , v) r + 1 2 d p (u 1 , u t ) r -cd p (v, u t ) r 󰁬 1 r ,
for r := max(p, 2), and c := c(p) ∈ (0, 1). Adding these estimates and using the triangle inequality we arrive at:

d p (u 0 , u 1 ) ≤ 󰁫 (t + ε) r d p (u 0 , u 1 ) r -cd p (v, u t ) r 󰁬 1 r + 󰁫 (1 -t + ε) r d p (u 0 , u 1 ) r -cd p (v, u t ) r 󰁬 1 r
After dividing by d p (u 0 , u 1 ), basic calculus yields that

d p (v, u t ) r d p (u 0 , u 1 ) r ≤ max 󰀕 (t + ε) r -t r c , ( 1
-t + ε) r -(1 -t) r c 󰀖 , implying that d p (v, u t ) ≤ ε 1 r
Cd p (u 0 , u 1 ), as desired.

4

The metric geometry of weak L p geodesic rays

For u ∈ E p ω let R p u denote the space of finite energy L p geodesic rays emanating from u. Note that we don't assume that the rays are unit speed, or even non-constant.

Following terminology from metric space theory [START_REF] Bridson | Metric spaces of non positive curvature[END_REF]

→ R p v . Moreover d p (u t , v t ) ≤ d p (u, v), t ≥ 0.
Proof. We first observe that the inequality d p (u t , v t ) ≤ d p (u, v) holds for two parallel rays {u t } t , {v t } t . Indeed, if {u t } t is parallel to {v t } t , then by (1) we have, for 0 < t < s,

d p (u t , v t ) ≤ 󰀕 1 - t s 󰀖 d p (u 0 , v 0 ) + t s d p (u s , v s ),
with the last term converging to 0 as s → +∞. Hence

d p (u t , v t ) ≤ d p (u, v), t ≥ 0. If {w t } t ∈ R p u is another ray which is parallel to {u t } t , then it is also parallel to {v t } t , hence d p (w t , v t ) ≤ d p (w 0 , v 0 ) = 0, giving the uniqueness.
We first argue the existence part of the proposition for u ≥ v, using the maximum principle. Consider the finite energy geodesic segments [0

, t] ∋ l → v t l ∈ E p ω , with v t 0 = v and v t t = u t .
Then by the comparison principle for geodesics we get that, for 0

≤ l ≤ t ≤ t ′ , v t ′ t ≤ u t = v t t , hence v t ′ l ≤ v t l . Also, (1) implies d p (v t l , u l ) t -l ≤ d p (v, u) t .
Putting the last two sentences together, [8, Proposition 4.3] implies that l → v l := lim t→∞ v t l ∈ E p ω is a finite energy geodesic ray such that d p (u l , v l ) ≤ d p (u, v), l ≥ 0. If u ≤ v, the proposition holds by the same argument (the inequality v t l ≤ v t ′ l being the only difference).

To treat the general case, we simply notice that h := max(sup X u, sup X v) ∈ H ω ⊂ E p ω and h ≥ u, v. By the above arguments, we can introduce a ray

{h t } t ∈ R p h such that d p (u t , h t ) ≤ d p (u, h). Since h ≥ v, it is now possible to introduce another ray {v t } t ∈ R p v with d p (v t , h t ) ≤ d p (v, h). The estimate d p (u t , v t ) ≤ d p (u, h) + d p (v, h), now follows from the triangle inequality, hence {v t } t is parallel to {u t }.
Next we introduce the chordal metric on R p u :

d c u,p ({u t } t , {v t } t ) := lim t→∞ d p (u t , v t ) t , {u t } t , {v t } t ∈ R p u . (19)
That the above increasing limit exists and is finite follows again from (1) and the triangle inequality. As we now clarify, (R p ω , d c p ) is in fact a complete geodesic metric space. Theorem 4.2. For any u ∈ E p ω , p ≥ 1, (R p u , d c u,p ) is a complete metric space. Moreover for any v ∈ E p ω the map

P uv : (R p u , d c u,p ) → (R p v , d c v,p
) is an isometry. Some aspects of the proof below can be traced back to [START_REF] Berman | Regularity of weak minimizers of the K-energy and applications to properness and K-stability[END_REF]Lemma 3.1].

Proof. That d c p,u satisfies the triangle inequality follows from the triangle inequality of d p . To argue non-degeneracy, suppose that d c p,u ({u t } t , {v t } t ) = 0. This implies that the increasing function f (t) = d p (u t , v t )/t satisfies f (0) = 0 and lim t→∞ f (t) = 0. Consequently f (t) = 0, t ≥ 0, implying that u t = v t , t ≥ 0. Now suppose that {u j t } t ⊂ R p u is a d c u,p -Cauchy sequence. Fixing l > 0 we have that

d p (u j l , u k l ) l ≤ d c u,p ({u j t } t , {u k t } t ). ( 20
)
Consequently The reverse inequality, also holds due to symmetry, showing that P uv is an isometry.

{u j l } j ⊂ E p ω is a d p -
By this theorem, no extra information is gained by choice of initial metric, hence going forward we will only consider the space (R p ω , d c p ), the collection of rays emanating from 0 ∈ H ω ⊂ E p ω .

Approximation of finite energy rays. In this paragraph we point out that bounded geodesic rays (running inside PSH(X, ω) ∩ L ∞ ) are dense among the rays of R p ω . Later, in the presence of finite radial K-energy we will sharpen this result further.

First we start with an auxilliary result, which is a consequence of Corollary 3.2, and it is the radial analog of [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Lemma 4.16]: Lemma 4.3. Let {u t } t , {u j t } ∈ R p ω such that u j t is decreasing (increasing a.e.) to u t as j → ∞ for all t ≥ 0. Then, d c p ({u j t } t , {u t } t ) → 0.

Proof. We start by noticing that t → sup X u t and t → sup X u j t are linear (Lemma 2.2). By our assumption we have that sup X u j 1 → sup X u 1 [56, Proposition 8.4], hence after possibly subtracting the same t-linear term from all our rays, without loss of generality we can assume that sup X u t , sup X u j t ≤ 0. By convexity we will obtain that 0 ≥ u j t ≥ u t (0 ≥ u t ≥ u j t ) for all j and t ≥ 0. Consequently, Corollary 3.2 is applicable to yield that:

d p (u j t , u t ) p t p ≤ |d p (0, u t ) p -d p (0, u j t ) p | t p = |d p (0, u 1 ) p -d p (0, u j 1 ) p |, t ≥ 0, (21)
where we have used that t → d p (0, u j t ) and t → d p (0, u t ) are linear. This is because u j 0 = u 0 = 0 and hence d p (u j t , 0) = td p (u j 1 , 0) as u j t is a geodesic ray. Now [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Lemma 4.16] gives that d p (u j 1 , u 1 ) → 0, in particular d p (0, u j 1 ) → d p (0, u 1 ), finishing the proof.

Remark 4.4. Analyzing the above argument we see that in Lemma 4.3 the conditions can be significantly weakened in some cases. For example, it is enough to assume that u t ≤ u j t , t ≥ 0, j ≥ 0, there exists C > 0 such that u j 1 ≤ C, j ≥ 0, and that u j 1 converges to u 1 pointwise on X, with the exception of a pluripolar set. Using [33, Lemma 5.1] we obtain that d p (u 1 , u j 1 ) p ≤ 󰁕 X |u 1u j 1 | p ω n u 1 , and the dominated convergence theorem allows to conclude that the right hand side of (21) still converges to zero. Theorem 4.5. Let {u t } t ∈ R p ω . Then there exists a sequence {u j t } t ∈ R p ω such that u j t ∈ PSH(X, ω) ∩ L ∞ and u j t ↘ u t as j → ∞ for all t ≥ 0. In particular d c p ({u j t } t , {u t } t ) → 0, and we can choose {u j t } t such that max 󰀓 u t , (sup

X u 1 -j)t 󰀔 ≤ u j t ≤ t sup X u 1 . (22)
Proof. It follows from Lemma 2.2 that t → sup X u t /t, t > 0 is constant, hence we can assume (by adding Ct to u t ) that sup X u t = 0, t ≥ 0. By convexity of t 󰀁 → u t (x), for x ∈ X fixed, we have

u t (x) ≤ l -t l -s u s (x) + t -s l -s u l (x) ≤ l -t l -s u s (x), 0 < s < t < l.
Letting l → +∞ we see that t → u t is t-decreasing. For τ ∈ R and x ∈ X we introduce

ψ τ (x) := inf t>0 (u t (x) -tτ ). (23)
From Kiselman's minimum principle [START_REF] Kiselman | The partial Legendre transformation for plurisubharmonic functions[END_REF] we have that ψ τ ≡ -∞ or ψ τ ∈ PSH(X, ω). More precisely, since sup X u t = 0 we have that ψ τ ∈ PSH(X, ω) for τ ≤ 0, and ψ τ ≡ -∞ for all τ > 0. Observe also that τ → ψ τ is τ -decreasing and τ -concave. For all x ∈ X with ψ 0 (x) > -∞ the curve t → u t (x) is continuous in (0, +∞). Hence, by the involution property of the Legendre transform, for such x we have

u t (x) = sup τ <0 (ψ τ (x) + tτ ) = sup τ ∈R (ψ τ (x) + tτ ), t > 0. ( 24
)
For ε > 0, τ < 0, set ψ ε τ (x) := max(0, 1 + ετ )ψ τ , and φ ε τ := P [ψ ε τ ].

We define φ ε 0 := lim τ →0 -φ ε τ . Since τ → ψ τ is τ -concave, τ -decreasing, and ψ τ ≤ 0, it is elementary to see that τ → ψ ε τ is also τ -concave and τ -decreasing. By elementary properties of P [•] we get that τ → φ ε τ is also τ -concave and τ -decreasing (see the proof of [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF]Proposition 4.6]). As a consequence of a result due to Ross-Witt Nyström [START_REF] Ross | Analytic test configurations and geodesic rays[END_REF] (further elaborated in [START_REF] Darvas | L 1 metric geometry of big cohomology classes[END_REF]

, Corollary 1.3]) the curve [0, ∞) ∋ t → u ε t (x) := sup τ <0 (φ ε τ (x) + tτ ) ∈ PSH(X, ω) ∩ L ∞ (25)
is a (bounded) geodesic ray emanating from 0. We now prove that u ε t ↘ u t as ε ↘ 0, for any t ≥ 0. For t = 0 there is nothing to prove since u ε 0 = u 0 = 0 on X. Fix now t > 0 and x ∈ X with ψ 0 (x) > -∞. Then, using τ -concavity, there exists C > 0 depending on ψ 0 (x), t (but not on ε) such that

u ε t (x) = sup -C≤τ ≤0
(φ ε τ (x) + tτ ), and u t (x) = sup -C≤τ ≤0

(ψ τ (x) + tτ ).

By Lemma 4.6 below, the family of functions τ 󰀁 → φ ε τ (x) decreases pointwise to the function τ 󰀁 → ψ τ (x) as ε → 0 + for τ < 0. Using τ -concavity and the fact that ψ 0 (x) > -∞, one can extend this convergence to τ = 0 as well. Hence by Dini's theorem the convergence is uniform on [-C, 0]. It thus follows that u ε t (x) ↘ u t (x) as ε → 0 + . We conclude that u ε t decreases to u t a.e. on X. But these are ω-psh functions, so the convergence holds everywhere on X.

That d c p ({u ε t } t , {u t } t ) → 0 as ε → 0 + , simply follows from Lemma 4.3. Since, φ ε τ = 0 for τ ≤ -1/ε and ψ τ ≤ φ ε τ , basic properties of Legendre transforms imply that u t ≤ u ε t ≤ 0 and -t ε ≤ u ε t ≤ 0, since ψ τ ≤ φ ε τ for all τ and φ ε τ = 0 for τ < -1 ε . This immediately yields [START_REF] Calabi | The space of Kähler metrics[END_REF] with ε = 1/j. Lemma 4.6. Assume that {u t } t ∈ R 1 ω satisfies sup X u t = 0 for all t ≥ 0. Then for ψ τ defined in [START_REF] Chen | The space of Kähler metrics[END_REF] we have that 󰁕 X ω n ψτ > 0 for all τ < 0. Additionally for any τ < 0,

lim ε→0 φ ε τ = lim ε→0 P [(1 + ετ )ψ τ ] = ψ τ . ( 26
)
Proof. By the involution property, application of the Legendre transform twice gives back the original convex function. In particular, we have that sup τ ψ τ (x) = lim τ →-∞ ψ τ (x) = u 0 (x) for all x ∈ X such that lim t→0 u t (x) = 0. In particular, we get that ψ τ increases a.e. to 0 as τ → -∞. According to [37, Remark 2.5] we obtain that lim

τ →-∞ 󰁕 X ω n ψτ = 󰁕 X ω n > 0.
Fixing τ < 0, this last identity implies existence of τ 0 < τ such that 󰁕 X ω n ψτ 0 > 0. By τ -concavity of τ → ψ τ we get that

ψ τ ≥ τ τ 0 ψ τ 0 + 󰀓 1 - τ τ 0 󰀔 ψ 0 .
Finally, by monotonicity [80, Theorem 1.2] and the multi-linearity of the non-pluripolar mass we obtain that 󰁕 X ω n ψτ > 0, as desired. To argue [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF], we start by noting that lim ε→0 P [(1 + ετ )ψ τ ] ≥ ψ τ , and according to [80, Theorem 1.2] and [37, Theorem 2.3 and Remark 2.5] we get that 󰁝

X ω n ψτ ≤ 󰁝 X ω n limε P [(1+ετ )ψτ ] ≤ lim ε 󰁝 X ω n P [(1+ετ )ψτ ] = lim ε→0 󰁝 X ω n (1+ετ )ψτ .
By multilinearity of the non-pluripolar Monge-Ampère product [15, Proposition 1.4] the last limit is 󰁕 X ω n ψτ . Hence we have equality everywhere, and all the integrals are positive. Consequently, lim ε→0 P [(1 + ετ )ψ τ ] ∈ F ψτ with the notation of [START_REF] Darvas | Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity[END_REF]Theorem 3.12].

It follows from [32, Proposition 5.1] (or [36, Lemma 3.17]) that P [ψ τ ] = ψ τ , for all τ ≤ 0 (the result in these works is only stated for rays of bounded potentials, however the proof only uses the comparison principle that holds for finite energy rays as well, implying the result for these more general rays). Putting everything together [37, Theorem 3.12] implies that lim ε→0 P [(1 + ετ )ψ τ ] ≤ P [ψ τ ] = ψ τ , as desired.

The construction of geodesic segments in R p ω .

Next we show that points of (R p ω , d c p ) can be connected by geodesic segments. We first treat the case p > 1, where due to uniform convexity, the construction can be carried out directly. The case p = 1 will be treated using approximation, via Theorem 4.5. The proof below shares similarities with the angle bisection techniques of [START_REF] Kleiner | Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings[END_REF].

Proof. By Theorem 4.2, we only have to show that any two rays {u t } t , {v t } t ∈ R p ω can be joined by a distinguished d c p -geodesic when p > 1. For any t ≥ 0, we denote by [0, 1] ∋ α → h t,α ∈ E p ω the finite energy geodesic connecting u t and v t . Then d p (u t , h t,α ) = d p (h t,0 , h t,α ) = d p (h t,0 , h t,1 ) = αd p (u t , v t ). To avoid introducing further variables, by [0, t] ∋ s → s t h t,α ∈ E p ω we denote the finite energy geodesic connecting 0 and h t,α . Finally, we can assume that u t ∕ = v t for t large enough. Indeed, if this does not hold, then (1) would give that {u t } t = {v t } t and the geodesic connecting the two rays is the constant one.

First we show that for any α ∈ [0, 1] and l ≥ 0 there exists w l,α ∈ E p ω such that lim t→∞ l t h t,α = w l,α . By endpoint stability of geodesic segments ([8, Proposition 4.3]), this will automatically imply that {w t,α } t ∈ R p ω . As we will see, α → {w t,α } t will represent the d c p -geodesic connecting {u t } t and {v t } t . Again, from (1) it follows that for any α ∈ [0, 1] and 0 < s ≤ t we have

d p (u s , s t h t,α ) s ≤ d p (u t , h t,α ) t = αd p (u t , v t ) t ≤ αd c p ({u t } t , {v t } t ), ( 27
)
d p (v s , s t h t,α ) s ≤ d p (v t , h t,α ) t = (1 -α)d p (u t , v t ) t ≤ (1 -α)d c p ({u t } t , {v t } t ), ( 28
)
where the identities in the middle follow from the fact that t → h t,α is a geodesic. We fix ε > 0. Since dp(us,vs) s 27) and ( 28) imply existence of s α,ε > 0 such that for any s α,ε ≤ s ≤ t we have

↗ d c p ({u t } t , {v t } t ), (
d p (u s , s t h t,α ) s ≤ (α + ε) d p (u s , v s ) s and d p (v s , s t h t,α ) s ≤ (1 -α + ε) d p (u s , v s ) s .
Now Proposition 3.6 implies that d p (h s,α , s t h t,α ) ≤ ε 1 r Cd p (u s , v s ) for any s α,ε ≤ s ≤ t. In particular, using (1), for any fixed l > 0 such that max(l, s α,ε ) ≤ s ≤ t we have

d p ( l s h s,α , l t h t,α ) l ≤ d p (h s,α , s t h t,α ) s ≤ ε 1 r C • d p (u s , v s ) s ≤ ε 1 r Cd c p ({u t } t , {v t } t ). ( 29
)
By shrinking ε, the expression on the right can be chosen to be as small as we want, implying that the sequence { l t h t,α } t ∈ E p ω is d p -Cauchy. This is the crucial step! By [33, Theorem 2], (E p ω , d p ) is complete, hence lim t l t h t,α =: w l,α ∈ E p ω , as proposed. Moreover, letting t → ∞ on the left hand side of ( 27) and ( 28), we obtain that

d p (u s , w s,α ) s ≤ αd c p ({u t } t , {v t } t ) and d p (v s , w s,α ) s ≤ (1 -α)d c p ({u t } t , {v t } t ), s > 0.
Letting s → ∞, together with the triangle we obtain

d c p ({u t } t , {v t } t ) ≤ d c p ({u t } t , {w t,α } t ) + d c p ({w t,α } t , {v t } t ) ≤ αd c p ({u t } t , {v t } t ) + (1 -α)αd c p ({u t } t , {v t } t ) = d c p ({u t } t , {v t } t ).
Thus everything is equal. In particular we have

d c p ({u t } t , {w t,α } t ) = αd c p ({u t } t , {v t } t ) and d c p ({w t,α } t , {v t } t ) = (1 -α)d c p ({u t } t , {v t } t ).
Suppose now that 0 ≤ α ≤ β ≤ 1. These last two identities together with the triangle inequality give that

(β -α)d c p ({u t } t , {v t } t ) ≤ d c p ({w t,β } t , {w t,α } t ). ( 30
)
To finish the proof we show that equality holds in this estimate. Indeed, another application of (1) gives that

d p ( l s h s,α , l s h s,β ) l ≤ d p (h s,α , h s,β ) s = (β -α)d p (u s , v s ) s , s > 0.
Letting s → ∞ in this estimate, and after that l → ∞, the reverse inequality in [START_REF] Chu | On the C 1,1 regularity of geodesics in the space of Kähler metrics[END_REF] follows, finishing the proof.

The

d c p -geodesic segment [0, 1] ∋ α → {w t,α } t ∈ R p
ω constructed in the above theorem will be called the d c p -chord joining {w t,0 } and {w t,1 }, as this curve is reminiscent of the chords joining the different points in the unit sphere of R n .

Finally, using approximation, we point out that the same result holds for p = 1 as well. First we remark that d c p -chords are automatically d c p ′ -chords for any p ′ ≤ p. This observation is of independent interest, and is the "radial version" of a well known phenomenon for the family of metric spaces (E p ω , d p ), p ≥ 1:

Proposition 4.8. Let 1 ≤ p ′ < p and {u t } t , {v t } t ∈ R p ω . Trivially {u t } t , {v t } t ∈ R p ′ ω , and the d c p -chord [0, 1] ∋ α → {w t,α } t ∈ R p ω connecting {u t } t , {v t } t is also a d c p ′ -chord.
Proof. To start, we trace the steps in the proof of Theorem 4.7 and notice that the curves α → h t,α , introduced in the argument, did not depend on the particular choice of p. Fixing l ≥ 0 and α ∈ [0, 1], the crux of the proof is the fact that d p 󰀃 l s h s,α , l s h s,α 󰀄 → 0 as s, t → ∞, which follows from uniform convexity (in case p > 1), as elaborated in [START_REF] Chen | Space of Kähler metrics (V)-Kähler quantization[END_REF]. Since 1 ≤ p ′ < p, we have that d p ′ (•, •) ≤ d p (•, •) and E p ω ⊂ E p ′ ω , hence the same conclusion holds for p ′ as well:

d p ′ 󰀓 l s h s,α , l t h t,α 󰀔 ≤ d p 󰀓 l s h s,α , l t h t,α 󰀔 → 0 as s, t → 0.
The rest of the proof does not use uniform convexity, and goes through without any difficulties for p ′ in place of p, arriving at the conclusion that the chord [0

, 1] ∋ α → {w t,α } t ∈ R p ω ⊂ R p ′ ω is a d c p ′ -chord as well. Theorem 4.9. (R 1 ω , d c 1
) is a complete geodesic metric space. Moreover, the d c 1 -chords of this space can be constructed by the method of Theorem 4.7.

Proof. Given {w t,0 } t , {w t,1 } t ∈ R 1 ω , we will show that there exists a

d c 1 -chord [0, 1] ∋ α 󰀁 → {w t,α } t ∈ R 1
ω joining {w t,0 } t and {w t,1 } t . Fix any p > 1. Using Theorem 4.5 we can find

{w k t,0 } t , {w k t,1 } t ∈ R p ω ⊂ R 1 ω such that w k t,0 ↘ w t,0 and w k t,1 ↘ w 1 t for all t ≥ 0. Let [0, 1] ∋ α → {w k t,α } t ∈ R p ω ⊂ R 1 ω be the d c
1 -geodesic joining {w k t,0 } t , {w k t,1 } t , which exists by Proposition 4.8. We look at the construction of the curves α → {w k t,α } in the proof of Theorem 4.7 and attempt to construct α → {w t,α } using the same method.

Using the fact that d 1 (u, v) = I(u) -I(v) for u ≥ v, and affinity of I along finite energy geodesics, one deduces that for any α ∈ [0, 1] and 0 ≤ s < t we have

d 1 󰀓 s t h k t,
󰀔 + d c 1 󰀓 s t h k t,α , w k s,α 󰀔 + d 1 (w k s,α , w s,α ).
Putting everything together, for s ≥ 0 fixed, the first and last term on the right hand side can be made arbitrarily small for big k. Next, with k fixed, the same is true for the middle term for big t, i.e., d 1 󰀃 s t h t,α , w s,α 󰀄 → 0 as t → ∞. As pointed out in the proof of Proposition 4.8, with this last fact in hand the rest of the proof of Theorem 4.7 goes through without any issues for p = 1.

Convexity of the radial K-energy.

Let p ≥ 1. The radial K-energy is defined for any {u t } t ∈ R p ω , and is given by the expression

K{u t } := lim t→∞ K(u t ) t ,
where K : E p ω → (-∞, ∞] is the extended K-energy of Mabuchi from [START_REF] Berman | Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties[END_REF][START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF]. In the setting of unit speed geodesics, this definition agrees with the ¥ invariant of [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF]. Also, there is clear parallel with the non-Archimedean K-energy of [START_REF] Boucksom | Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF] (and references therein).

Lemma 4.10. Let {u t } t ∈ R p u , {v t } t ∈ R p v parallel, with u, v ∈ E p satisfying K(u) < +∞ and K(v) < +∞. Then K{u t } = K{v t }.
Proof. By the proof of Proposition 4.1 we can assume that either u ≤ v or v ≤ u.

For each t > 0 let [0, t] ∋ l 󰀁 → v t l ∈ E p ω be the finite energy geodesic connecting v t 0 := v and v t t := u t . It follows from Proposition 4.1 (and its proof) that lim t→+∞ d p (v t l , v l ) = 0 for each l fixed. By convexity of K [8, Theorem 1.2], for any 0 < l < t we have that

K(v t l ) ≤ 󰀓 1 - l t 󰀔 K(v) + l t K(u t ).
Thus, letting t → +∞ and using lower semicontinuity of K w.r.t. d p [8, Theorem 4.7] we obtain

K(v l ) l ≤ K(v) l + K{u t }.
Letting l → +∞ yields K{v t } ≤ K{u t }. The reverse inequality is obtained by reversing the roles of u, v.

By the above lemma it makes sense to restrict to R p ω when considering the radial K-energy. Since d c p -convergence implies d c 1 -convergence it follows from [27, Proposition 5.9] that the resulting functional

K{•} : R p ω → (-∞, ∞] is d c p -lsc.
In the last result of this section we point out that K{•} is also convex along the chords of R p ω for any p ≥ 1: Theorem 4.11. Suppose that p ≥ 1 and

[0, 1] ∋ α → {w t,α } t ∈ R p ω is a d p -chord joining {u t } t , {v t } t ∈ R p ω . Then α → K{w t,α } is convex. Proof.
We use the notation and terminology of the proof of Theorems 4.7 and 4.9, and normalize K such that K(0) = 0. Using convexity of K along finite energy geodesics [8, Theorem 1.2] we know that for any 0 < s ≤ t and α ∈ [0, 1] we have

K 󰀃 s t h t,α 󰀄 s ≤ K(h t,α ) t ≤ (1 -α) K(u t ) t + α K(v t ) t . Since d p ( s t h t,α , w s,α ) → 0, given that K is d p -lsc ([8, Theorem 1.2]) it follows that K(w s,α ) s ≤ lim inf t→∞ K 󰀃 s t h t,α 󰀄 s ≤ (1 -α)K{u t } + αK{v t }.
The result now follows after taking the limit s → ∞.

Remark 4.12. Many theorems that hold for the finite energy metric spaces (E p ω , d p ) admit a radial version for (R p ω , d p ). As we already pointed out, Theorem 1.4, Lemma 4.3, and also Theorem 5.1 below are examples of this phenomenon. This does not seem to be limited to only these results either. Indeed, though we will not pursue this further here, one can introduce radial analogs of the operators max(•, •) and P (•, •), and similar identities/inequalities/results hold for these as the ones described in [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF][START_REF] Darvas | The Mabuchi completion of the space of Kähler potentials[END_REF].

Approximation with converging radial K-energy

Approximation with rays of bounded potentials

The goal of this subsection is to strengthen the conclusion of Theorem 4.5 and obtain Theorem 1.5(i) in the process:

Theorem 5.1. Let {u t } t ∈ R p ω , p ≥ 1.
Then there exists a sequence {u j t } t ∈ R ∞ ω such that u j t decreases to u t , for each t > 0 fixed and K{u j t } → K{u t }. In particular lim j→+∞ d c p ({u j t }, {u t }) = 0. In addition, sup X u j t = sup X u t for all j, t > 0. In case K{u t } = +∞, by the fact that K{•} is d c p -lsc [27, Proposition 5.9], we will simply invoke Theorem 4.5 for the existence of the sequence of {u j t } t . If K{u t } is finite, we will need a much more delicate argument, resting on the relative Ko󰀀 lodziej type estimate of [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]Theorem 3.3], as detailed in the argument below.

At places, the argument below shares some similarities with the proof of [41, Theorem 3.2], with the relative Ko󰀀 lodziej type estimate of [START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF] taking the place of Perelman's estimates along the Kähler-Ricci flow on Fano manifolds. Before engaging in the proof of Theorem 5.1, we prove an auxiliary lemma:

Lemma 5.2. Let {u t } t ∈ R 1 ω with sup X u t = 0, t ≥ 0. Then lim j→∞ lim sup t→+∞ 󰁝 {ut≤-jt} (-u t ) t ω n ut = 0. (32)
Proof. It follows from Theorem 4.5 that we can choose {u j t } t ∈ R 1 ω such that u t ≤ max(u t , -jt) ≤ u j t ≤ 0 and d c 1 ({u j t }, {u t }) = 0. As u j t ≥ u t , we have that d 1 (u j t , u t ) = I(u j t ) -I(u t ) [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Corollary 4.14]. Hence lim t→+∞ I(u j t )-I(ut) t = 0. From monotonicity and elementary properties of I(•) we conclude that

lim t→∞ I(max(u t , -jt)) -I(u t ) t = 0, ultimately implying 0 ≤ lim j lim t→∞ 󰁝 X max(u t , -jt) -u t t ω n ut ≤ (n + 1) lim j lim t→∞ I(max(u t , -jt)) -I(u t ) t = 0.
Consequently both limits are equal to zero, and on the set {u t ≤ -2jt}, we have 0 ≥ max(u t , -jt)u t ≥ -ut 2 . This and the above together yield [START_REF] Darvas | Weak geodesic rays in the space of Kähler potentials and the class E(X, ω)[END_REF]. Proof of Theorem 5.1. Using Theorem 4.5 and the fact that K{•} is d c p -lsc, we can assume that K{u t } < +∞. Also, via Lemma 2.2, by possibly adding Ct to u t we can additionally assume that sup X u t = 0, i.e., t → u t is t-decreasing with u ∞ := lim t→∞ u t ∈ PSH(X, ω).

For each j > 1, l > 1, we let ϕ j l ∈ E(X, ω) be the unique ω-psh function, whose existence is guaranteed by [55, Theorem A], such that

ω n ϕ j l = 󰀕 1 - 1 2 j 󰀖 {u l >-jl} ω n u l + a j,l ω n , sup X ϕ j l = 0, ( 33
)
where 0 ≤ a j,l ≤ 1 is a constant arranged so that the measure on the right hand side has total mass equal to 󰁕 X ω n . Next we point out that the conditions of Theorem 2.3 are satisfied with appropriate choice of data. Let a := 󰀃 1 -

1 2 j 󰀄 1/2 ∈ (0, 1), u := ϕ j l , χ := 󰀃 1 -1 2 j
󰀄 1/2n max(u l , -jl), and f := 1. Then, using locality of the non-pluripolar complex Monge-Ampère measure (see e.g. [55, Corollary 1.7]) we have that ω n max(u l ,-jl) ≥ {u l >-jl} ω n u l , hence,

ω n u ≤ aω n χ + f ω n .
Moreover, due to [15, Proposition 4.3] and [39, Lemma 4.2], there exists A(X, ω) > 0 such that for any Borel set E ⊂ X we have

󰁝 E f ω n = 󰁝 E ω n ≤ ACap ω (E) 2 ≤ A 󰀕 1 - 󰀕 1 - 1 2 j 󰀖 1/2n 󰀖 -2n Cap χ (E) 2 ,
where Cap ω is the usual Monge-Ampère capacity and Cap χ is its relative version from [39, Section 3]. Lastly, we note that χ ≤ 0 = P [ϕ j l ], due to [32, Theorem 3], hence all the conditions of Theorem 2.3 are satisfied to imply that

ϕ j l = u ≥ χ -C j ≥ max(u l , -jl) -C j , (34)
where C j > 0 is a constant depending on j, but not l > 1! In particular ϕ j l is bounded.

Moreover, for 1 < j < k and l > 1 we have

ω n ϕ j l ≤ 1 -2 -j 1 -2 -k ω n ϕ k l + ω n .
Similarly to [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF], this allows for another application of Theorem 2.3, with the choice of data a

:= 󰀃 1-2 -j 1-2 -k 󰀄 1/2 ∈ (0, 1), u := ϕ j l , χ := 󰀃 1-2 -j 1-2 -k
󰀄 1/2n ϕ k l , and f := 1. Similarly to the above, the conditions of Theorem 2.3 are satisfied to yield that

ϕ j l = u ≥ χ -C j,k ≥ ϕ k l -C j,k , (35)
where C j,k > 0 depends on j, k, but not on l > 1! For each l > 1 let [0, l] ∋ t 󰀁 → u j,l t be the bounded geodesic segment joining 0 and ϕ j l + C j . Then ( 34) and ( 35) together with the comparison principle for finite energy geodesics implies that

C j t l ≥ u j,l t ≥ max(u t , -jt), t ∈ [0, l], (36)
and

C j t l ≥ u j,l t ≥ u k,l t - D j,k t l , 0 < j < k, t ∈ [0, l], (37)
where D j,k depends on j, k but not on l > 1.

To show that the above geodesic sequences subconverge to appropriate geodesic rays, we first prove a number of estimates in the claims below. Claim 1. For any j > 1 we have lim sup

t→+∞ Ent 󰀃 ω n , ω n ϕ j t 󰀄 t ≤ lim sup t→+∞ Ent(ω n , ω n ut ) t .
Since Ent(ω n , ω n ut ) < +∞ (because K(u t ) < +∞), for any t ≥ 0, we can write ω n

ϕ j t = f t,j ω n
and ω n ut = f t ω n . Observe first that for any g t ≥ 0 with

󰁕 X g t ω n = 󰁕 X ω n we have that lim sup t→+∞ 󰁝 X g t log(g t ) t ω n = lim sup t→+∞ 󰁝 X (g t + B) log(g t + B) t ω n , ∀B ≥ 1.
This follows after splitting up both integrals using the partition {0 ≤ g t ≤ C} and {C < g t } for C > 0 big and noticing that the lim sup of integrals on {0 ≤ g t ≤ C} is always zero. By construction, see [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF], 1 ≤ f t,j + 1 ≤ f t + 2 and hence, since s 󰀁 → s log(s), s > 1 is increasing, (f t,j + 1) log(f t,j + 1) ≤ (f t + 2) log(f t + 2). Using the above we then conclude:

lim sup t→+∞ 󰁝 X f t,j log(f t,j ) t ω n = lim sup t→+∞ 󰁝 X (f t,j + 1) log(f t,j + 1) t ω n ≤ lim sup t→+∞ 󰁝 X (f t + 2) log(f t + 2) t ω n = lim sup t→+∞ 󰁝 X f t log f t t ω n .
Claim 2. We have

lim j lim sup t→+∞ I(ϕ j t , u t ) t = 0.
Before we start with the argument, we recall that

I(v, w) = 󰁕 X (v -w)(ω n w -ω n v ) for v, w ∈ E 1
ω . By [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF] we have

I(ϕ j t , u t ) ≤ 1 2 j 󰁝 X |ϕ t j -u t |ω n ut + 󰁝 {ut≤-jt} |ϕ t j -u t |ω n ut + 󰁝 X |ϕ t j -u t |ω n ,
and the claim follows from the following three estimates. First, the estimate of ( 34) and basic properties of I(•) give that lim

j lim sup t→+∞ 1 2 j 󰁝 X |ϕ t j -u t | t ω n ut ≤ lim j lim sup t→+∞ 1 2 j C j t + lim j lim sup t→+∞ 1 2 j |I(u t )| t = 0. (38)
Second, by the dominated convergence theorem we have that

lim t→+∞ 󰁝 X |ϕ j t -u t | t ω n ≤ lim t→+∞ 󰁝 X |C j -u ∞ | t ω n = 0. (39)
Ent 󰀓 ω n , ω n ϕ j l k 󰀔 -Ent 󰀓 ω n , ω n u l k 󰀔 l k ≤ lim sup k→+∞ Ent 󰀓 ω n , ω n ϕ j l k 󰀔 l k -lim k Ent 󰀓 ω n , ω n u l k 󰀔 l k = lim sup k→+∞ Ent 󰀓 ω n , ω n ϕ j l k 󰀔 l k -lim sup t→+∞ Ent 󰀃 ω n , ω n ut 󰀄 t ≤ lim sup t→+∞ Ent 󰀓 ω n , ω n ϕ j t 󰀔 t -lim sup t→+∞ Ent 󰀃 ω n , ω n ut 󰀄 t .
It thus follows from Claim 1 that

lim j→+∞ lim sup k→+∞ Ent 󰀓 ω n , ω n ϕ j l k 󰀔 -Ent 󰀓 ω n , ω n u l k 󰀔 l k ≤ 0. ( 41
)
We continue with

lim sup k→+∞ K(ϕ j l k ) l k ≤ lim sup k→+∞ K(ϕ j l k ) -K(u l k ) l k + lim sup k→+∞ K(u l k ) l k .
Thus, using the Chen-Tian formula (8) together with [START_REF] Darvas | Geodesic rays and Kähler-Ricci trajectories on Fano manifolds[END_REF] and the estimates of Claims 3, 4, we can continue to write that

lim j→+∞ lim sup k→+∞ K(ϕ j l k ) l k ≤ K{u t }.
As a result, there exists an increasing sequence {j m } m ⊂ N such that

lim sup k→+∞ K(ϕ jm l k ) l k ≤ K{u t } + 1 m .
Hence, returning to the geodesic segments constructed at the beginning of the argument, by convexity of the K-energy we have, for all t ∈ [0, l k ],

lim sup k→+∞ K(u jm,l k t ) t ≤ lim sup k→+∞ K(ϕ jm l k ) l k ≤ K{u t } + 1 m . ( 42
)
Let us fix m ≥ 1 and t ∈ Q + momentarily. We use the compactness property of E 1 ω (see [START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF]Corollary 4.8]) to extract a subsequence (again denoted by

l k = l k (m, t)) such that d 1 (u jm,l k t , u m t ) → 0 as k → ∞ for some u m t ∈ E 1 ω .
Using a diagonal Cantor process it is actually possible to pick the same subsequence of {l k } k for each m ≥ 1 and t ∈ Q + . Moreover, due to the endpoint stability of geodesic segments [8, Proposition 4.3], we get that the convergence extends for all t ≥ 0: there exists u m t ∈ E 1 ω such that d 1 (u jm,l k t , u m t ) → 0 as k → ∞ for any t ≥ 0 and {u m t } t ∈ R ∞ ω . Now we prove additional properties for our sequence {u m t } t . By [START_REF] Darvas | On the singularity type of full mass currents in big cohomology classes[END_REF], we notice that

{u m t } t ∈ R ∞ ω : max(u t , -j m t) ≤ u m t ≤ 0. ( 43
)
Moreover, by [START_REF] Darvas | Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity[END_REF] we also have that {u m t } t is m-decreasing! Fixing t > 0, since d 1 (u jm,l k t , u m t ) → 0 as k → ∞, due to d 1 -lower semicontinuity of K, from [START_REF] Darvas | Weak geodesics in the space of Kähler metrics[END_REF] we obtain that

K(u m t ) t ≤ K{u t } + 1 m , ∀t > 0, (44)
hence K{u m t } ≤ K{u t } + 1 m , as desired. Next, we argue that d 1 (u m t , u t ) → 0 for any t ≥ 0, as m → ∞. But this is simply a consequence of Claim 3. Indeed, due to [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF], we only need to argue that:

lim m→+∞ d 1 (u m t , u t ) t = lim m→+∞ I(u m t ) -I(u t ) t = 0. ( 45
)
But from I-linearity, for any t ∈ [0, l k ] we have that [START_REF] Darvas | Quantization in geometric pluripotential theory[END_REF] together with u t ≤ u m t implies the normalization sup X u m t = sup X u t = 0, t ∈ [0, 1], as desired.

I(u m t ) -I(u t ) t =
In the above argument we have used the following lemma whose proof goes along the same lines as [7, Theorem 5.8]: Lemma 5.3. There exists a continuous non-decreasing function f : R + → R + with f (0) = 0 such that for all 0

≥ ϕ 1 , ϕ 2 , ψ ∈ E 1 ω , we have 󰀏 󰀏 󰀏 󰀏 󰁝 X (ϕ 1 -ϕ 2 )(ω n ϕ 2 -ω n ψ ) 󰀏 󰀏 󰀏 󰀏 ≤ Af (I(ϕ 1 , ϕ 2 )/A),
where A = max(-I(ϕ 1 ), -I(ϕ 2 ), -I(ψ), 1).

In the proof below we use C n > 0 to denote various numerical constants (only dependent on dim X = n) and f : R + → R + to denote a continuous non-decreasing function such that f (0) = 0. They may be different from place to place.

Proof. By approximation of finite energy potentials from above by smooth ones, we can assume that ϕ 1 , ϕ 2 , ψ are smooth (the convergence of the integrals is assured by the results of [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF], see for example [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Proposition 2.11]). We set u = ϕ 1ϕ 2 and v = (ϕ 1 + ϕ 2 )/2. For p = 0, ..., n let

a p := 󰁝 X uω p ϕ 2 ∧ ω n-p ψ and b p := 󰁝 X i∂u ∧ ∂u ∧ ω p v ∧ ω n-p-1 ψ .
It follows from [55, Proposition 2.5] that

I(ψ 1 , ψ 2 ) ≤ C n (|I(ψ 1 )| + |I(ψ 2 )|), for all 0 ≥ ψ 1 , ψ 2 ∈ E 1 ω . (46)
In particular, I(ψ, ϕ j ) ≤ C n A, j = 1, 2. For p = 0, 1, ..., n -1 we have, using integration by parts and by the Cauchy-Schwarz inequality,

|a p -a p+1 | = 󰀏 󰀏 󰀏 󰀏 󰁝 X i∂u ∧ ∂(ψ -ϕ 2 ) ∧ ω p ϕ 2 ∧ ω n-p-1 ψ 󰀏 󰀏 󰀏 󰀏 ≤ 󰀕󰁝 X i∂u ∧ ∂u ∧ ω p ϕ 2 ∧ ω n-p-1 ψ 󰀖 1 2 󰀕󰁝 X i∂(ψ -ϕ 2 ) ∧ ∂(ψ -ϕ 2 ) ∧ ω p ϕ 2 ∧ ω n-p-1 ψ 󰀖 1 2 ≤ C n b 1 2 p (I(ϕ 2 , ψ)) 1/2 ≤ C n b 1/2 p A 1 2 .
In the last line above we have used ω ϕ 2 ≤ 2ω v and the inequality 󰁝

X i∂(ψ -ϕ 2 ) ∧ ∂(ψ -ϕ 2 ) ∧ ω p ϕ 2 ∧ ω n-p-1 ψ ≤ 󰁝 X (ψ -ϕ 2 )(ω n ϕ 2 -ω n ψ ).
It thus follows, by summing up the estimates of |a pa p+1 | above for p = 0, ..., n -1, that

|a 0 -a n | ≤ C n A 1 2 n-1 󰁛 p=0 b 1 2 p . (47)
We claim that there is a non-decreasing continuous function f :

R + → R + with f (0) = 0 such that b p ≤ Af (I(ϕ 1 , ϕ 2 )/A), 0 ≤ p ≤ n -1.
We proceed by (backwards) induction. For p = n -1 we can simply take f (s) = C n s, s ≥ 0. By the same argument as above using integration by parts and the Cauchy-Schwarz inequality we have, for 0

≤ p ≤ n -2, b p -b p+1 = 󰁝 X i∂u ∧ ∂u ∧ i∂ ∂(ψ -v) ∧ ω p v ∧ ω n-p-2 ψ ≤ 󰀏 󰀏 󰀏 󰀏 󰁝 X i∂u ∧ ∂(ψ -v) ∧ i∂ ∂u ∧ ω p v ∧ ω n-p-2 ψ 󰀏 󰀏 󰀏 󰀏 ≤ 󰀏 󰀏 󰀏 󰀏 󰁝 X i∂u ∧ ∂(ψ -v) ∧ (ω ϕ 1 -ω ϕ 2 ) ∧ ω p v ∧ ω n-p-2 ψ 󰀏 󰀏 󰀏 󰀏 ≤ 󰀏 󰀏 󰀏 󰀏 󰁝 X i∂u ∧ ∂(ψ -v) ∧ ω ϕ 1 ∧ ω p v ∧ ω n-p-2 ψ 󰀏 󰀏 󰀏 󰀏 + 󰀏 󰀏 󰀏 󰀏 󰁝 X i∂u ∧ ∂(ψ -v) ∧ ω ϕ 2 ∧ ω p v ∧ ω n-p-2 ψ 󰀏 󰀏 󰀏 󰀏 ≤ C n 󰀕󰁝 X i∂u ∧ ∂u ∧ ω p+1 v ∧ ω n-p-2 ψ 󰀖 1 2 󰀕󰁝 X i∂(ψ -v) ∧ ∂(ψ -v) ∧ ω p+1 v ∧ ω n-p-2 ψ 󰀖 1 2 ≤ C n I(ψ, v) 1 2 b 1 2 p+1 ,
where we used several times that ω ϕ j ≤ 2ω v . Using [START_REF] Demailly | Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Actes du Colloque en l'honneur de P. Dolbeault[END_REF] we thus have

b p ≤ b p+1 + AC n (b p+1 /A) 1 2 ≤ Af (I(ϕ 1 , ϕ 2 )/A) + AC n f (I(ϕ 1 , ϕ 2 )/A) 1 2 .
Consequently, by possibly increasing f , we have that b p ≤ Af (I(ϕ 1 , ϕ 1 )/A), proving the claim. Comparing with (47), we thus have

|a 0 -a n | ≤ Af (I(ϕ 1 , ϕ 2 )/A),
what we wanted to prove.

Approximation with rays of C 1, 1 potentials

The goal of this subsection is to prove Theorem 1.5(ii):

Theorem 5.4. Let p ≥ 1. Suppose that {u t } t ∈ R p ω is such that K{u t } < ∞. Then there exists {v k t } t ⊂ R 1, 1 ω such that v k t ↘ u t , t ≥ 0, d c p ({v k t } t , {u t } t ) → 0 and K{v k t } → K{u t }. Additionally, sup X v k t = sup X u t for all k, t > 0.
To argue this result, we need two auxiliary theorems, whose proof will be given at the end of the section. First we will need the following theorem, which will allow to obtain "scaled" C 1, 1 estimates along geodesic rays, via convexity:

Theorem 5.5. Let [0, 1] ∋ t → u t ∈ H 1, 1 ω be the C 1, 1-geodesic connecting u 0 , u 1 ∈ H 1, 1 ω .
Then there exists B > 0, only depending on (X, ω) such that

[0, 1] ∋ t → ess sup X (log(n + ∆ ω u t ) -Bu t ) ∈ R is a convex function.
The proof of this theorem is obtained using the estimates developed in [START_REF] He | On the space of Kähler potentials[END_REF]. We will also need the following smoothing argument along bounded geodesic rays, relying on the regularizing property of the weak Monge-Ampère flows, closely following the arguments of [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF]: Theorem 5.6. Let B > 0 be from Theorem 5.5, and {u t } t ∈ R ∞ ω with K{u t } < ∞ and sup X u t = 0, t ≥ 0. Then there exists α > 0 (depending on {u t } t ) such that for all s > 0 and j ∈ N one can find u j s ∈ H ω satisfying the following conditions: (i) {u j s } j is decreasing and

u s ≤ u j s ≤ αj2 -j , (ii) sup X 󰀃 log(n + ∆ ω u j s ) -Bu j s 󰀄 ≤ α2 j (1 + s), (iii) d 1 (u j s , u s ) ≤ α2 -j s + αj2 -j , (iv) Ent(ω n , ω n u j s ) ≤ Ent(ω n , ω n us
). Proof of Theorem 5.4. First we assume that {u t } t ∈ R ∞ ω and sup X u t = 0, t ≥ 0. If {u t } t is the constant ray then we are done, hence after rescaling we can also assume that d 1 (0, u t ) = t, t ≥ 0. Let {u j s } s>0,j∈N be the potentials constructed as in Theorem 5.6. Let us fix j ∈ N momentarily. Given s > 0, by [0, s] ∋ t → u j,s t ∈ H 1, 1 ω we denote the C 1, 1 geodesic connecting u j,s 0 := 0 and u j,s s := u j s . Using condition (i) in Theorem 5.6 and the comparison principle for weak geodesics we get that

u t ≤ u j,s t ≤ αj2 -j t s , t ∈ [0, s]. (48)
Since {u j s } j is decreasing, by the comparison principle for weak geodesics, {u j,s t } j is decreasing as well, for any t ∈ [0, s].

Given t ∈ (0, s], by condition (ii) in Theorem 5.6 and Theorem 5.5 we have that ess sup X (log(1

+ 1 n ∆ ω u j,s t ) -Bu j,s t ) t ≤ sup X (log(1 + 1 n ∆ ω u j s ) -Bu j s ) s ≤ α2 j 󰀓 1 + 1 s 󰀔 . (49)
Finally, (1) and condition (iii) in Theorem 5.6 implies that

d 1 (u j,s t , u t ) t ≤ α2 -j + α s , t ∈ [0, s]. (50)
Fixing t > 0, [START_REF] Dervan | The Kähler-Ricci flow and optimal degenerations[END_REF] and [START_REF] Nezza | Geometry and Topology of the space of Kähler metrics on singular varieties[END_REF] gives that {u j,s t } {s>t} is compact in the C 1,α topology, implying existence of v j t ∈ H 1, 1 ω such that 󰀂v j tu j,s t 󰀂 C 1,α → 0 as s → ∞ (after passing to an ssubsequence). Moreover, letting s → ∞ in (48), ( 49) and (50), using Lemma 7.1, we arrive at

u t ≤ v j t ≤ 0, 1 t 󰀓 log(1 + 1 n ∆ ω v j t ) -Bv j t 󰀔 ≤ α2 j , d 1 (v j t , u t ) t ≤ α2 -j , t ∈ (0, ∞). (51)
Using an Arzela-Ascoli type argument exactly the same way as in the proof of Theorem 5.1, after passing to another s-subsequence, we can assume that 󰀂v j tu j,s t 󰀂 C 1,α → 0 for all t > 0 at the same time, implying existence of {v j t } ∈ R 1, 1 for any j ∈ N. By (51) we get that d c 1 ({v j t }, {u t }) → 0 as j → ∞. Remark 4.4 implies that d c p ({v j t }, {u t }) → 0, as desired.

Finally, since {u j,s t } j is decreasing for any t ∈ [0, s], a diagonal Cantor process now implies that {v j t } j can be chosen to be decreasing for any t > 0. To show that K{v j t } → K{u t }, we first note that by [27, Proposition 5.9] we have K{u t } ≤ lim inf j K{v j t }. Hence it is enough to show that K{v j t } ≤ K{u t } + f (α2 -j ) for any j ∈ N, where f : R + → R + is some continuous function with f (0) = 0.

Using conditions (iii) and (iv) in Theorem 5.6 we can start writing:

K(u j s ) -K(u s ) s ≤ 󰀏 󰀏 S 󰀏 󰀏 d 1 (u j s , u s ) s + n I Ric(ω) (u j s ) -I Ric(ω) (u s ) s ≤ α s + α2 -j 󰀏 󰀏 S 󰀏 󰀏 + n I Ric(ω) (u j s ) -I Ric(ω) (u s ) s . (52)
We can suppose that -Cω ≤ Ric(ω) ≤ Cω, and for the rest of the proof C > 0 will denote a constant only dependent on (X, ω). Using condition (i) in Theorem 5.6 multiple times, we can start the following estimates

I Ric(ω) (u j s ) -I Ric(ω) (u s ) s ≤ C 󰁓 j 󰁕 X (u j s -u s )ω ∧ ω j u j s ∧ ω n-j-1 us s ≤ C 󰁓 j 󰁕 X (u j s -u s )ω n u j s /4+us/4 s ≤ C 󰁓 j 󰁕 X (u j s -u s )(ω n u j s /4+us/4 -ω n us ) s + C I(u j s ) -I(u s ) s ≤ f (I(u j s , u s )/s) + Cα2 -j + Cα s ,
where f : R + → R + is a continuous function with f (0) = 0, and in the last line we used Lemma 5.3. Together with [START_REF] Donaldson | Scalar curvature and stability of toric varities[END_REF], this inequality implies that

K(u j,s t ) t ≤ K(u j s ) s ≤ K{u t } + Cα2 -j + f (I(u j s , u s )/s) + Cα s .
Letting s → ∞, since K is convex and d 1 -lsc, we obtain that K{v j t } ≤ K{u t } + Cα2 -j + f (α2 -j ), as desired, finishing the proof when {u

t } t ∈ R ∞ ω . Now let {u t } t ∈ R p ω with K{u t } < ∞.
We can still assume that sup X u t = 0. By Theorem 5.1, there exists

{u k t } ∈ R ∞ ω such that u k t ↘ u t , t ≥ 0, d c p ({u k t }, {u t }) ≤ 1 2 k , 󰀏 󰀏 K{u t } -K{u k t } 󰀏 󰀏 ≤ 1 2 k and sup X u k t = 0. Let {u k,j
s } j be the potentials of Theorem 5.6 associated to the rays {u k t } t . By the construction of these potentials (as elaborated in the proof of Theorem 5.6) and [56, Corollary 2.2] it follows that {u k,j s } k,j is decreasing both in j and k for any fixed s > 0. Using this, a diagonal Cantor process applied to the simultaneous approximation of each {u k t } t described above, yields rays Remark 5.7. It follows from [START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF], that the approximating rays {v j t } t ∈ R 1, 1 in the previous theorem additionally satisfy the estimate:

{v k t } t ∈ R 1, 1 ω such that u k t ≤ v k t , d c p ({v k t }, {u k t }) ≤ 1 2 k , 󰀏 󰀏 K{u k t } -K{v k t } 󰀏 󰀏 ≤ 1 2 k , moreover (
1 t ess sup X 󰀓 log(1 + 1 n ∆ ω v j t ) -Bv j t 󰀔 ≤ α2 j , t > 0, j ∈ N.
The proof of Theorem 5.5. Let us denote the log of the left hand side of (6) by F (u ε ). Given a smooth function h ∈ C ∞ ([0, 1]×X), if h attains its maximum at (t, x) ∈ (0, 1)×X, then ellipticity of (6) gives that

DF (u ε )(h)(t, x) := d ds 󰀏 󰀏 󰀏 s=0 F (u ε + sh)(t, x) ≤ 0. ( 53
)
Proof of Theorem 5.5. Let us first assume that u 0 , u 1 ∈ H ω and let [0, 1] ∋ t 󰀁 → u ε t be the smooth ε-geodesic connecting u 0 , u 1 (see Section 2.1). Fix (t, x) ∈ [0, 1] × X and ε > 0 momentarily. In [59, page 339] (after equation (2.19)) it is shown that for some constants B, C > 1, dependent only on (X, ω), we have that

DF (u ε )(log(n + ∆ ω u ε t ) -Bu ε t )(t, x) ≥ n 󰁛 j=1 1 1 + (u ε t ) j j -C, (54)
The proof of Theorem 5.6. In the proof of Theorem 5.6 we will use the formalism of [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF] adapted to our context. Fixing ϕ 0 ∈ E 1 ω with sup X ϕ 0 = 0, we consider the following parabolic PDE on [0, ∞) × X with initial data given by ϕ 0 :

d dt ϕ t = log 󰀗 (ω + i∂ ∂ϕ t ) n ω n 󰀘 . (58)
To avoid cumbersome notation, we will denote t-derivatives by dots throughout this paragraph. As shown in [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF], (t, x) → ϕ t (x) is smooth on (0, ∞) × X. The initial condition simply means that d 1 (ϕ t , ϕ 0 ) → 0, as t → 0 [56, Section 5.2.2]. In case ϕ 0 ∈ H ω , we actually have that 󰀂ϕ tϕ 0 󰀂 C ∞ → 0 as t → 0. Moreover it is shown in [50, Theorem B] that if ϕ 0 ∈ E 1 ω and ϕ j 0 ∈ H ω converges in L 1 (X, ω n ) to ϕ 0 , then for any t > 0 we have that 󰀂ϕ j tϕ t 󰀂 C ∞ → 0, where {ϕ j t } j are the smooth solutions to [START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF] with initial data ϕ j 0 . All this implies that the apriori estimates and maximum principles developed in [56, Section 2] for smooth initial data, also apply for initial data in E 1 ω , as above (for our applications ϕ 0 will be actually bounded).

For the remainder of this paragraph we pick a small constant λ > 0 depending only on (X, ω) such that 󰁕 X e -2λφ ω n is uniformly bounded for all φ ∈ PSH(X, ω) normalized by sup X φ = 0 (see [75, Proposition 2.1], [START_REF] Zeriahi | Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions[END_REF]).

Let v be the unique continuous ω-psh function such that ω n v := e λv-λϕ 0 -n log λ ω n .

By our choice of λ, it follows from [START_REF] Ko󰀀 | The complex Monge-Ampère equation[END_REF][START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF] (or much more generally [39, Theorem 5.3]) that v is uniformly bounded by a constant depending only on (X, ω). This implies that l → ψ t is a subsolution to [START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF], and an application of the maximum principle [56, Corollary 2.2] yields the first inequality in [START_REF] He | On Calabi's extremal metric and properness[END_REF].

The second inequality follows from [56, Corollary 2.2], after comparing t → ϕ t with the constant solution t → χ t := sup X ϕ 0 = 0 of [START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF].

Simplifying [START_REF] He | On Calabi's extremal metric and properness[END_REF], we actually obtain that:

ϕ 0 ≤ ϕ t + Ct -Ct log t, t ∈ [0, 1], (61)
for some constant C > 0 dependent on (X, ω). This can be taken one step further, as we now describe:

Corollary 5.9. There exists a constant C > 1 depending on (X, ω) such that w t ≥ w t/2 for any t ∈ [0, 1], where w t := ϕ t + Ct -Ct log t.

Proof. Fixing s ∈ (0, 1), we apply [START_REF] Kell | Uniformly convex metric spaces[END_REF] to the flow t 󰀁 → ϕ s/2+t , starting from ϕ s/2 . By [START_REF] He | On Calabi's extremal metric and properness[END_REF] and [START_REF] Kell | Uniformly convex metric spaces[END_REF] we have that 󰀂e -λϕ s/2 󰀂 L 2 is controlled by 󰀂e -λϕ 0 󰀂 L 2 which is uniformly bounded by a constant depending on (X, ω). Hence for t := s/2 ∈ [0, 1] in [START_REF] Kell | Uniformly convex metric spaces[END_REF] In the second line above we have used φt = ∆ ωϕ t φt which follows from the equation of the flow [START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF]. Consequently, t → Ent(ω n , ω n ϕt ) is decreasing on [0, ∞), when ϕ 0 ∈ H ω . For general ϕ 0 ∈ E 1 ω , let ϕ j 0 ∈ H ω be such that d 1 (ϕ j 0 , ϕ 0 ) → 0 and Ent(ω n , ω n For the remainder of this paragraph, let {u t } t ∈ R ∞ with sup X u t = 0, t ≥ 0 and K{u t } < +∞, as in the statement of Theorem 5.6. Since sup X u s = 0, s ≥ 0, by the weak L 1 -compactness of PSH(X, ω) we have that u s ↘ u ∞ ∈ PSH(X, ω).

We fix s > 0 for the remainder of this paragraph, and we construct the sequence u j s as follows. For each j we define u j s := w s,2 -j , where w s,t ∈ H ω is constructed in [START_REF] Kiselman | The partial Legendre transformation for plurisubharmonic functions[END_REF] with respect to the flow t → ϕ s,t , starting from ϕ s,0 := u s . The estimate of Corollary 5.9, together with (61) yields the condition (i) in

Proof. From [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF]Corollary 4.5] we obtain that for any j ∈ N and s > 0 we have 1 2 j log(n + ∆ ω u j s ) = 1 2 j log(n + ∆ ω ϕ s,2 -j ) ≤ C(osc X ϕ s,2 -j-1 + 1), [START_REF] Ko󰀀 | The complex Monge-Ampère equation[END_REF] where C > 0 only depends on (X, ω). Using [START_REF] Kell | Uniformly convex metric spaces[END_REF] we have that osc X ϕ s,2 -j-1 ≤inf X ϕ s,0 + α =inf X u s + α.

By [START_REF] Darvas | Weak geodesic rays in the space of Kähler potentials and the class E(X, ω)[END_REF]Theorem 1] we have that inf X u s = m {ut} s for some constant m {ut} ≤ 0. Consequently (63) follows after putting the above together with condition (i) in Theorem 5.6 (and possibly increasing the value of α > 0).

Next we address condition (iii) in Theorem 5.6:

Lemma 5.12. We have that d 1 (u j s , u s ) ≤ α2 -j s + αj2 -j for any j ∈ N, s > 0.

Proof. For the flow t 󰀁 → ϕ s,t , using the equation ( 58), we can write where we have used Lemma 5.10. Recall that for u j s := w s,2 -j , due to property (i) we can continue: d 1 (u j s , u s ) = I(u j s ) -I(u s ) ≤ I(ϕ s,2 -j ) -I(ϕ s,0 ) + αj2 -j ≤ Ent(ω n , ω n us )2 -j + αj2 -j .

After invoking Lemma 5.13 below, and possibly adjusting α > 0 again, the proof is finished.

As promised above, we argue that along {u t } t the entropy has sublinear growth: Lemma 5.13. There exists C > 0 depending on d 1 (0, u 1 ) and K{u t } such that Ent(ω n , ω n ut ) ≤ Ct, t ≥ 0.

Proof. Let D > K{u t }. By the Chen-Tian formula for the extended K-energy [START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF] gives the bound n|I Ric(ω) (u t )| ≤ C 2 d 1 (0, u t ). Since u t is a geodesic ray starting from 0, we have that d 1 (u t , 0) = td 1 (0, u 1 ). This gives the last estimate above. The constants C 1 , C 2 , C 3 depend only on an upper bound for K{u t }, d 1 (0, u 1 ), and (X, ω).

Proof. Fix g ∈ G. Using [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF]Lemma 5.8], and the fact that (g -1 ) * 󰀃 g * ω 󰀄 = ω, we obtain that 0 = g -1 .(g.0) = g -1 .0 + (g -1 ) * 󰀃 g.0 󰀄 . In particular, we have that inf X g.0 = sup X g -1 .0.

Due to [START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF]Corollary 4] and [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Lemma 3.45] we have that 0 ≤ sup

Now we address the Laplacian estimate. To start, we note that there exists C := C(X, ω) > 0 such that -Cω ≤ Ric(ω) ≤ Cω. Pulling back by g we obtain that Ric(ω g.0 ) ≤ Cω g.0 . We introduce F g := log

󰀔

. We obtain that i∂ ∂F g = Ric(ω) -Ric(ω g.0 ) ≥ -Cω -Cω g.0 .

In particular, 1 2 g.0 + 1 2C F g ∈ PSH(X, ω), implying that sup

X 󰀕 1 2 g.0 + 1 2C F g 󰀖 ≤ C + 󰁝 X 󰀕 1 2 g.0 + 1 2C F g 󰀖 ω n ≤ 1 2 d 1 (0, g.0) + C.
Here, we used Jensen's inequality to obtain 󰁕 X F g ω n ≤ 0. Using (73) we arrive at: sup

  Proof. To begin, let p ≥ 1 and λ ∈ [0, 1]. By density (and [8, Proposition 4.3]) we can assume

Theorem 4 . 7 .

 47 If p > 1, then (R p ω , d cp ) is a complete geodesic metric space.

Lemma 5 . 8 .

 58 With λ ∈ (0, 1) and v as above, we have that(1λt)ϕ 0 + λtv + n(t log tt) ≤ ϕ t ≤ 0, t ∈ [0, 1]. (60)Proof. Let ψ t := (1λt)ϕ 0 + λtv + n(t log tt), t ∈ [0, 1]. The following hold:ψt = λ(vϕ 0 ) + n log t = log 󰀕 λ n t n • ω

Ent(ω n , ω n ϕ j 0 )

 0 n , ω n ϕ 0 ) (such sequence exists by[START_REF] Berman | Convexity of the extended K-energy and the large time behavior of the weak Calabi flow[END_REF] Theorem 1.3]). Fixing t > 0, by[START_REF] Di Nezza | Uniqueness and short time regularity of the weak Kähler-Ricci flow[END_REF] Theorem B] we have that ϕ j t → C ∞ ϕ t , hence we can conclude that Ent 󰀃 ω n , ω n = Ent(ω n , ω n ϕ 0 ), finishing the proof.

  Theorem 5.6 for α := 3C. Condition (iv) follows automatically from Lemma 5.10. Next we address condition (ii) in Theorem 5.6, which is closely related to [56, Corollary 4.5]: Lemma 5.11. We have that sup X 󰀃 log(n + ∆ ω u j s ) -Bu j s 󰀄 ≤ α2 j (1 + s), j ∈ N, s > 0.

I

  (ϕ s,t ) -I(ϕ s,0 ) = 󰁝 t 0 d dl I(ϕ s,l )dl = 󰁝 t 0 Ent(ω n , ω n ϕ s,l )dl ≤ Ent(ω n , ω n ϕ s,0 )t = Ent(ω n , ω n us )t,

g - 1

 1 ω n + C ≤ Cd 1 (0, g.0) + C, .0 ω n ≤ Cd 1 (0, g -1 .0) + C.Since d 1 (0, g -1 .0) = d 1 (g.0, 0), putting the above together with[START_REF] Semmes | Complex Monge-Ampère and symplectic manifolds[END_REF], one of the desired estimates follows: sup X |g.0| ≤ Cd 1 (0, g.0) + C.

XF󰀃

  g ≤ Cd 1 (0, g.0) + C.[START_REF] Székelyhidi | Extremal metrics and K-stability[END_REF] To obtain the Laplacian estimate, we start with Yau's calculation (for a survey, see[START_REF] Boucksom | An introduction to the Kähler-Ricci flow[END_REF] Proposition 4.1.2]):Tr ω g.0 󰀅 i∂ ∂ log Tr ω ω g.0 󰀆 ≥ Tr ω 󰀅 i∂ ∂ log 󰀃 ω n g.0 ω n 󰀄󰀆 Tr ω ω g.0 -CTr ω g.0 ω,where C > 0 only depends on (X, ω). Let B := 2C + 1. Using the fact that Ric(ω g.0 ) ≤ Cω g.0 , we can continue:Tr ω g.0 󰀅 i∂ ∂󰀃 log Tr ω ω g.0 -Bg.0 󰀄󰀆 ≥ Tr ω 󰀅 -Cω g.0 -Cω 󰀆 Tr ω ω g.0 -CTr ω g.0 ω -BTr ω g.0 󰀅 i∂ ∂g.0 󰀆 ≥ -nC Tr ω ω g.0 + (B -C)Tr ω g.0 ω -nB -C ≥ (B -2C)Tr ω g.0 ω -nB -C ≥ Tr ω g.0 ω -C ≥ 󰀓 ω Tr ω ω g.0 󰀄 1 n-1 -C.Let x 0 ∈ X be the point where 󰀃 log Tr ω ω g.0 -Bg.0 󰀄 is maximized. Using the above estimate and (74) we obtain that Tr ω ω g.0 (x 0 ) ≤ Cd 1 (0.g.0) + C. Together with[START_REF] Song | Bergman metrics and geodesics in the space of Kähler metrics on toric varieties[END_REF] we arrive at sup X log(n + ∆ ω (g.0)) ≤ Cd 1 (0, g.0) + C.

  K is G-invariant and there exists δ > 0 s.t. K{u t } ≥ δd 1 (0, u 1 ) for all G-calibrated geodesic rays {u t } t ∈ R 1, 1.

			1,	1
	and L 1 version of the uniform geodesic stability conjecture are equivalent. As alluded to
	previously, the breakthrough of Chen-Cheng [26, 27] together with [35, Theorem 4.7] es-
	sentially yielded the L 1 version of this conjecture (see Theorem 6.1 below). Putting things
	together, we arrive at our most important main result, essentially settling Conjecture 1.7:
	Theorem 1.8 (C 1, 1 uniform geodesic stability). Let (X, ω) be a compact Kähler manifold.
	Then the following are equivalent:
	(i) There exists a csck metric in H. (ii) There exists δ > 0 such that K{u t } ≥ δ lim sup t (iii)	d 1,G (G0,Gut) t	for all {u t } t ∈ R 1, 1.

  and let h t ∈ E p ω be the d p -midpoint of the finite energy geodesic connecting u t , v t , t ∈ [0, 1]. Assuming that u t

  , two rays {u t } t , {v t } t are parallel if d p (u t , v t ) is uniformly bounded. Given the characteristics of the finite energy spaces, any ray admits a unique parallel ray emanating from an outside point, thus the d p -geometries verify Euclid's 5th postulate for half-lines, answering an open question raised in [27, Remark 1.6]: Proposition 4.1. Let u, v ∈ E p ω then for any {u t } t ∈ R p u there exists a unique {v t } t ∈ R p v such that {u t } t is parallel to {v t } t , giving a bijection P uv : R p u

  Cauchy sequence with limit u l ∈ E p ω . By the endpoint stability of geodesic segments in E p ω ([8, Proposition 4.3]) it follows that t → u t is a geodesic ray. More importantly, letting k → ∞ in (20) it follows that dp(u j l ,u l ) l is arbitrarily small for high enough j and any l > 0. This in turn implies that d c u,p ({u j t } t , {u t } t ) → 0, giving completeness. Let {u t } t , {h t } t ∈ R p u and let {v t } t := P uv ({u t } t ), {k t } t := P uv ({h t }. By the triangle inequality and Proposition 4.1 we have d u,p ({u t } t , {h t } t ) = lim

t→+∞ d p (u t , h t ) t ≤ lim t→+∞ d p (u t , v t ) + d p (v t , k t ) + d p (k t , h t ) t ≤ lim t→+∞ d p (u, v) + d p (v t , k t ) + d p (v, u) t = d u,p ({v t } t , {k t } t ).

  } k ∈ E 1 ω is decreasing and d 1 -bounded, hence by[START_REF] Darvas | The Mabuchi geometry of finite energy classes[END_REF] Lemma 4.16] there exists {w s,α } ∈ E 1 ω such that d 1 (w k s,α , w s,α ) → 0 as k → ∞. Lastly, the triangle inequality gives:

	α ,	s t	h t,α	󰀔	= I = s 󰀓 s t t I 󰀃 h k h k t,α t,α 󰀄 󰀔 --I s t I 󰀓 s t 󰀃 h t,α h t,α 󰀄 󰀔	(31)
					=	s(1 -α) t	(I(w k t,0 ) -I(w t,0 )) +	sα t	(I(w k t,1 ) -I(w t,1 ))
					= s(1 -α)(I(w k 1,0 ) -I(w 1,0 )) + sα(I(w k 1,1 ) -I(w 1,1 )),
	with the last expression converging to zero regardless of the values of t > 0. From here we get that d 1 󰀃 s t h k t,α , s t h t,α 󰀄 → 0 as k → ∞, uniformly with respect to t. On the other hand, by Proposition 4.8 (and its proof) we get that d c 1 ( s t h k t,α , w k s,α ) → 0 as t → ∞ for any fixed k ≥ 0. By construction, each sequence {w k s,α d 1 󰀓 s t h t,α , w s,α 󰀔 ≤ d 1 󰀓 s t h k t,α , s t h t,α

  This claim follows from Claim 2 and Lemma 5.3 below, with ϕ 1 = ϕ j t , ϕ 2 = u t and ψ = 0. Indeed, given[START_REF] Darvas | The Mabuchi completion of the space of Kähler potentials[END_REF], we have that max(-I(ϕ j t ), -I(u t )) ≃ Ct + C j , for a uniform constant C. Since α can be written as the difference of two Kähler forms, and I α (•) is monotone when α ≥ 0, notice that the claim follows if we can

	Using (34) we observe that this last identity is a consequence of
		lim j	lim sup t→+∞	I ω (ϕ j t + C j ) -I ω (u t ) t	= lim j	lim sup t→+∞	I ω (ϕ j t ) -I ω (u t ) t	= 0.
	However we have that
		I ω (ϕ j t ) -I ω (u t ) t	=	(n + 1)(I(ϕ j t ) -I(u t )) t	-	󰁕	X ϕ j t ω n ϕ j t	-t	󰁕	X u t ω n ut	.
	So, by Claim 3, it is enough to show that
					lim j	lim sup t→+∞	󰀏 󰀏	󰁕	X ϕ j t ω n ϕ j t	-t	󰁕	X u t ω n ut	󰀏 󰀏	= 0.
	Again, due to (34) and elementary properties of I(•) we have that
	Third, by Lemma 5.2 and (34), lim j lim sup t→+∞ 󰁝 {ut≤-jt} |ϕ j t -u t | t lim j lim sup t→+∞ 󰀏 󰀏 󰁕 X (ϕ j t -u t )ω n ω n ut ≤ lim j ϕ j t 󰀏 󰀏 t ≤ lim lim sup t→+∞ j lim sup 󰁝 t→+∞ {ut≤-jt} I(ϕ j t ) -I(u t ) |u t | + C j t t where the last identity follows from Claim 3. Due to (33) and (39) we also have ω n ut = 0. = 0, Claim 3. We have lim j lim sup t→+∞ |I(ϕ j t ) -I(u t )| t = 0. Lemma 5.3 then gives 󰀏 󰀏 󰀏 󰀏 󰁝 X (ϕ j t -u t )(ω n ut -ω n ) 󰀏 󰀏 󰀏 󰀏 ≤ (Ct + C j )f 󰀃 I(ϕ j t , u t )/(Ct + C j ) 󰀄 . lim j lim sup t→+∞ 󰀏 󰀏 󰀏 󰁕 X u t 󰀓 ω n ϕ j t -ω n ut 󰀔󰀏 󰀏 󰀏 t ≤ lim j lim sup t→+∞ 󰀕 1 2 j 󰁝 X |u t | t ω n ut + 󰁝 |u t | ω n ut t {ut≤-jt} where the last equality follows from Lemma 5.2 and the fact that 󰁕 X |u t |ω n ut ≤ d 1 (0, u t ) = (40) 󰀖 = 0, td 1 (0, u 1 ) ([33, Theorem 3]). Conclusion. There exists a sequence l k ↗ +∞ such that lim k→+∞ Ent 󰀃 ω n , ω n u l k 󰀄 /l k = lim sup t→+∞ Ent 󰀃 ω n , ω n ut 󰀄 /t. It then follows that
	Hence lim sup k→+∞				lim j	lim sup t→+∞	󰀏 󰀏 󰀏 󰀏	󰁝	X	(ϕ j t -u t ) t	(ω n ut -ω n )	󰀏 󰀏 󰀏 󰀏 = 0.
	Again, due to (34) and elementary properties of I(•) we have that
				0 ≤ lim sup t→+∞	I(ϕ j t ) -I(u t ) t	≤ lim sup t→+∞	󰁝	X	ϕ j t -u t t	ω n ut .
	Putting these last two estimates together and (39), the claim follows.
	Claim 4. For any closed smooth real (1, 1)-form α we have
						lim j	lim sup t→+∞	|I α (ϕ j t ) -I α (u t )| t	= 0.
	Recall that I α (v) := v . argue that 󰁓 n-1 j=0 󰁕 X vα ∧ ω j
	lim j	lim sup t→+∞	|I ω (ϕ j t + C j ) -I ω (u t )| t	= lim j	lim sup t→+∞	I ω (ϕ j t + C j ) -I ω (u t ) t	= 0.

  we have ϕ s ≥ ϕ s/2 -Cs/2 + C(s/2) log(s/2), where C only depends on (X, ω). Thus, after possibly increasing C > 0, the functionw t := ϕ t + Ct -Ct log t satisfies w t ≥ w t/2 , t ∈ [0, 1].We also point out the following simple monotonicity result:Lemma 5.10. The map [0, ∞) ∋ t → Ent(ω n , ω n ϕt ) ∈ R is decreasing.Proof. First let us assume that ϕ 0 ∈ H ω in[START_REF] Jost | Nonpositive curvature: geometric and analytic aspects[END_REF]. For t ≥ 0, we can start by computing

	d dt	Ent(ω n , ω n ϕt ) =	d dt 󰁝	󰁝	X	φt (ω + i∂ ∂ϕ t ) n
		=

X φt (ω + i∂ ∂ϕ t ) n -󰁝 X |∇ φt | 2 ωϕ t (ω + i∂ ∂ϕ t ) n = 󰁝 X 󰀃 ∆ ωϕ t φt 󰀄 (ω + i∂ ∂ϕ t ) n -󰁝 X |∇ φt | 2 ωϕ t (ω + i∂ ∂ϕ t ) n = -󰁝 X |∇ φt | 2 ωϕ t (ω + i∂ ∂ϕ t ) n ≤ 0.

  we obtain that Ent(ω n , ω n ut ) ≤ Dt -SI(u t ) + nI Ric(ω) (u t ) ≤ C 1 t + C 2 d 1 (0, u t ) + C 3 d 1 (0, u t ) ≤ Ct,where we have used[41, Proposition 2.5] in the second estimate. Here C 1 , C 2 , C 3 are uniform constants. Since u t ≤ 0 we have d 1 (0, u t ) = |I(u t )|, while [41, Proposition 2.5]

where we have used normal coordinates of ω at x and i∂ ∂u ε t is assumed to be diagonal. Additionally, fix δ > 0 and g δ (t) := δt 2 /2. We also have

Assume that h ε,δ (t, x) := log(n + ∆ ω u ε t ) -Bu ε t + g δ (t) is maximized at (t, x) ∈ (0, 1) × X. Then by [START_REF] Donaldson | Scalar curvature and projective embeddings, II[END_REF], [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF] and [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF] we obtain at (t, x) that

In the second line we have used the inequality of arithmetic and geometric means while for the last identity we have used the equation [START_REF] Berman | A variational approach to the Yau-Tian-Donaldson conjecture[END_REF]. Thus, for ε < δ(n + 1) n+1 /C n+1 we get a contradiction in the above inequality, implying that the maximum of h ε,δ can not be attained at (t, x), an interior point of [0, 1] × X. In particular, we have that sup

Letting ε ↘ 0 and δ ↘ 0 thereafter, via Lemma 7.1 we arrive at ess sup X h 0,0 (t, x) ≤ max 󰀃 sup

motivating the introduction of M u 0 ,u 1 (t) := ess sup X (log(n + ∆ ω (u t )) -Bu t ). Indeed, we can simply write

Next we observe that (57) also holds in case we merely have u 0 , u 1 ∈ H

6 Applications to geodesic stability

First we point out how the L 1 version of Conjecture 1.7 can be derived from [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF] and [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Theorem 4.7]. As alluded to in the introduction, the argument is implicitly contained in [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF][START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF], but we provide a short proof here as this result is not explicitly stated in that paper. Recall that G = Aut 0 (X, J), and for the definition of G-calibrated rays we refer back to the introduction. Theorem 6.1 (L 1 uniform geodesic stability). Let (X, ω) be a compact Kähler manifold.

Then the following are equivalent: (i) There exists a csck metric in H ω .

(ii) There exists δ > 0 such that

1 ω normalized by the condition I(u t ) = 0, t ≥ 0.

Proof. By [26, Theorem 1.5], the conditions of [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Theorem 4.7] are satisfied. Indeed, it was pointed out in [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF]Theorem 10.1] that all the conditions (A1)-(A4) and (P1)-(P6) hold with the exception of (P3), which is exactly the content of [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF]Theorem 1.5].

After comparing with the conclusion of [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Theorem 4.7], we only need to argue that condition (ii) implies that K is G-invariant. However we notice that (ii) implies that (X, ω) is L 1 -geodesically semistable, in the sense that, K{u t } ≥ 0 for any

To show that Theorem 1.8 holds, we argue in the next two results that conditions (ii) and (iii) in the previous theorem are equivalent with their C 1, 1 version: Theorem 6.2. Let (X, ω) be a compact Kähler manifold. Then the following are equivalent:

Proof. We only need to argue that (ii)⇒(i). Let {u t } t ∈ R 1 . We can assume that K{u t } < ∞, otherwise there is nothing to prove. We pick

Since the last term converges to zero as k → ∞, we obtain that lim sup t d 1,G (G0, Gu k t )/t converges to lim sup t d 1,G (G0, Gu t )/t, as desired. Theorem 6.3. Let (X, ω) be a compact Kähler manifold. Then the following are equivalent: (i) K is G-invariant and there exists δ > 0 s.t. K{u t } ≥ δd 1 (0, u 1 ) for all G-calibrated geodesic rays {u t } t ∈ R 1 . (ii) K is G-invariant and there exists δ > 0 s.t. K{u t } ≥ δd 1 (0, u 1 ) for all G-calibrated geodesic rays {u t } t ∈ R 1, 1.

Proof. We only need to argue that (ii)⇒(i). Let {u t } t ∈ R 1 , G-calibrated and nonconstant. We can assume that K{u t } < ∞, otherwise there is nothing to prove.

Using Theorem 5.4, we pick {u k t } t ∈ R

The following estimates will be used later:

where in the first line we have used the triangle inequality; in the second line we have used the definition of d 1,G and the fact that G acts on E 1 0 by d 1 -isometry (see [44, Lemma 5.9]); in the third line we have used the triangle inequality and that {u t } t is G-calibrated; in the last line we have used that (0, +∞) ∋ t 󰀁 → d 1 (u k t , u t )/t is increasing (see [START_REF] Berman | Regularity of weak minimizers of the K-energy and applications to properness and K-stability[END_REF]). Let [0, t] ∋ l → ρ k,t l ∈ E 1 ω be the finite energy geodesic connecting 0 and g k t .u k t . From (65) and [START_REF] Darvas | Geometric pluripotential theory on Kähler manifolds[END_REF]Lemma 4.9] it follows that

Using G-invariance, convexity of K, and that K(0) = 0, for any l ∈ [0, t] we have that

Due to [8, Corollary 4.8], after possibly selecting a subsequence t j → ∞, there exists ũk l ∈ E 1 ω for any l > 0, such that d 1 (ũ k l , ρ k,t j l ) → 0. After taking the limit in (67), due to [8, Proposition 4.3] we find that {ũ k t } t ∈ R 1 is G-calibrated. Moreover, due to (66), there exists k 0 such that {ũ k t } t is not the constant ray for k ≥ k 0 . Next we argue that {ũ k t } t ∈ R 1, 1. To start, for t ≥ 1 using (65) and the fact that G acts by d 1 -isometries (see [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF]Lemma 5.9]), we get that

Next, let B > 0 as in the statement of Theorem 5.5. Using Lemma 6.4 below, we have the following estimates max

Recall that g k t .u k t = (g k t ) * u k t + g k t .0 (see [START_REF] Darvas | Tian's properness conjectures and Finsler geometry of the space of Kähler metrics[END_REF]Lemma 5.8]). In particular, [32, Theorem 1], Remark 5.7 and (69) give that sup

where C depends on k but not on t ≥ 1! Using [32, Theorem 1] and Theorem 5.5, we find that sup

Due to the fact that K is d 1 -lsc, G-invariant and convex, similar to (68), we find that for all l > 0 and k ≥ k 0 we have:

≤ lim inf

where in the second line we have used [START_REF] Li | Special test configurations and K-stability of Fano varieties[END_REF], and all the denominators are non-zero since {ũ k t } t and {u k t } t are non-constant for k ≥ k 0 . Finally, we use that (ii) holds for {ũ k t } t ∈ R 1, 1. Consequently, after letting l, t → ∞ in [START_REF] Phong | The Monge-Ampère operator and geodesics in the space of Kähler potentials[END_REF], we arrive at

.

Letting k → ∞, we now obtain that δ ≤ K{ut} d 1 (0,u 1 ) , finishing the proof. Lemma 6.4. Let (X, ω) be a compact Kähler manifold. There exists C := C(X, ω) > 0 such that for all g ∈ G we have that sup X |g.0| ≤ Cd 1 (0, g.0) + C and sup X log(n + ∆ ω (g.0)) ≤ Cd 1 (0, g.0) + C.

The Laplacian estimate from this lemma is equivalent with the following estimate for the gradient ∇g, as a self map of X:

ω ≤ e Cd 1 (0,g.0)+C , g ∈ G.

The desired Laplacian estimate of the lemma can be extracted from the arguments of [START_REF] Chen | Compactness of Kähler metrics with bounds on Ricci curvature and I functional[END_REF], as we now elaborate.

Appendix

Here we address two likely known facts about Kähler potentials with bounded Laplacian, whose proof we could not find in the literature.

Proof. After picking subsequence, we can assume without loss of generality that the lim inf on the left hand side is actually a limit. Let δ ∈ R such that log(n+∆ ω u j (x))-Bu j (x) < δ for a.e. x ∈ X and j ∈ N. To conclude, it is enough to show that log(n + ∆ ω u) -Bu ≤ δ, a.e. on X.

By assumption, ∆ ω u j + n ≤ e Bu j +δ in the weak sense of postive measures on X. By Dini's lemma we have that 󰀂u j -u󰀂 C 0 → 0, hence passing to the weak limit we have that ∆ ω u + n ≤ e Bu+δ , again in the weak sense of positive measures on X. Since all our measures have bounded densities, (76) follows.

Complementing the above lemma, in the next result we point out that the quantity on the right hand side of ( 75) can be realized with an approriate decreasing sequence, constructed via the method of [START_REF] Demailly | Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Actes du Colloque en l'honneur de P. Dolbeault[END_REF]. Let us recall some elements of this work. We denote by exph x : T x X → X the "quasiholomorpic exponential map" of ω (see [START_REF] Demailly | Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Actes du Colloque en l'honneur de P. Dolbeault[END_REF]Section 2]). Let χ : R → R be an even non-negative smooth function supported in [0, 1] such that 󰁕 C n χ(󰀂ξ󰀂 2 )dλ(ξ) = 1. Given u ∈ PSH(X, ω), one can introduce u ε ∈ C ∞ (X) by the following formula: where dλ is the Lebesgue measure on T x X with respect to ω.

Proposition 7.2. Let u ∈ H 1, 1 ω and B ∈ R. There exists u j ∈ H ω such that u j converges to u decreasingly (and uniformly by Dini's lemma) and lim j sup X (log(n + ∆ ω u j ) -Bu j ) = ess sup X (log(n + ∆ ω u) -Bu).

(77)

Proof. By possibly rescaling u with a small constant, we can assume that there exists δ > 0 such that ω u ≥ δω. In particular, it follows from the estimate of [46, Theorem 4.1] that for small enough ε > 0 we actually have that u ε ∈ H ω . Moreover, 󰀂u ε -u󰀂 C 0 → 0. Also, it follows from [46, Theorem After possibly adding small constants to u ε , we can construct the decreasing sequence desired.