Clara L Aldana University

O F Luxembourg

Gilles Carron
email: gilles.carron@univ-nantes.fr

Universit É De Nantes

Samuel Tapie
email: samuel.tapie@univ-nantes.fr

A ∞ WEIGHTS AND COMPACTNESS OF CONFORMAL METRICS UNDER L n/2 CURVATURE BOUNDS

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension n, assuming uniform volume bounds and L n/2 bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gromov-Hausdorff topology. Our study is based on the use of A∞-weights from harmonic analysis, and provides geometric controls on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on R n with infinite volume and finite L n/2 norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

INTRODUCTION

This paper is devoted to the question of sequential compactness of Riemannian metrics inside a conformal class given uniform bounds on their volume and integral bounds on their curvatures. It was shown by Gursky in [START_REF] Gursky | Compactness of conformal metrics with integral bounds on curvature[END_REF] that (unless g 0 is conformal to the round sphere), if the Riemann curvature tensor is bounded in L p for some p > n 2 and if the volume is uniformly controlled, then the sequence of metric is pre-compact in the C α topology for some α ∈ (0, 1).

It was later shown that no such compactness can hold when the curvature tensor is bounded in L n/2 . Indeed, in [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF] Chang, Gursky and Wolff constructed two examples of families of sequences of conformal metrics with uniform volume and L n/2 curvature bounds which have no convergent sub-sequences. Let us briefly recall these examples.

Example 1: spherical blowup. Let (S n , g S ) be the standard sphere, N ∈ S n be some fixed point and S be its antipodal point. Let σ : S n \{N } → R n be the stereographic projection with pole N , and let eucl denote the Euclidean metric on R n . For all λ > 0, let g λ = σ * (λ eucl): it is well known that g λ extends to a smooth metric on S n , conformally equivalent to g S .

Note that since g λ is merely a pullback of g S by some conformal diffeomorphisms, the volume of (S n , g λ ) and all L p norms of the curvature tensor are preserved. Moreover, as λ → +∞, the Riemannian measures dµ g λ converge to a Dirac mass at S.

It should also be noticed that for any R > 0, the g S -ball B(S, R) of radius R contains asymptotically all the curvature of g λ . In particular, we have

lim λ→+∞ ˆB(S,R) (Scal g λ ) n 2 dµ g λ 2 n
= α(n, 2),

1
where α(n, 2) = σ 2 n

n n(n -1) is the L n/2 -total scalar curvature of the standard round unit sphere, and σ n = 2π n/2 Γ(n/2) is its n-th volume.

It was shown in [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF] that this family of conformal deformations of the sphere can be glued on any compact Riemannian manifold (M, g 0 ) at any point x 0 , giving rise to a sequence of conformal deformations (g λ = e 2f λ g 0 ) λ>0 such that:

• the volume and diameter of the complement of any open ball centered at x 0 go to 0 as λ → +∞; • the L n/2 norm of the Riemann curvature tensor are uniformly bounded;

• the sequence of Riemannian measures dµ g λ develop a Dirac mass at x 0 and the volume stays in some fixed interval [v, V ].

In particular, this implies that the associated sequence of distances (d g λ ) λ>0 is not precompact for the uniform C 0 topology. Once again, in this example, for any fixed R > 0, the g 0 -ball B(x 0 , R) of radius R contains asymptotically a large part of the scalar curvature of g λ . In particular it can be shown that, in this case, we have

lim inf λ→+∞ ˆB(x0,R) Scal + g λ n 2 dµ g λ 2 n
≥ α(n, 2).

We will see below that if we consider sequences of conformal measures whose volume and L n/2 -total scalar curvature are uniformly bounded, and the scalar curvature concentration (in L n/2 norm) in a g 0 -ball is strictly less than α(n, 2), then this kind of spherical blowup cannot occur.

Example 2: Schrödinger type blowup. The second construction in [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF] exploits the Schrödinger-type structure of the conformal Laplacian, where the potential is only in L n/2 . We present a simple example inspired by their construction.

Let (M, g 0 ) be a compact smooth Riemannian manifold, let x 0 ∈ M and R 0 be such that inj x0 (M, g 0 ) ≥ 2R 0 , where inj x0 (M, g 0 ) denotes the injectivity radius of g 0 at x 0 . For x ∈ M , define [START_REF] Akutagawa | Yamabe metrics of positive scalar curvature and conformally flat manifolds[END_REF] f

∞ (x) =    ln d(x0,x) R0 if 0 < d(x 0 , x) ≤ R 0 0 if d(x 0 , x) ≥ 2R 0
with uniformly bounded second derivatives when R 0 ≤ d(x 0 , x) ≤ 2R 0 . Let (f k : M → R) k∈N be a sequence of smooth maps which converges to f ∞ in W 2, n 2 . We consider for all k ≥ 0 the conformal metric g k = e 2f k g 0 . The volume of (M, g k ) is uniformly bounded, and it follows from Theorem 1.159 in [4, p 58] that the L n 2 norm of the Riemannian curvature of (M, g k ) are uniformly bounded. Note that eventhough the sequence of Riemannian metrics (g k ) will blow up at x 0 , the diameter of (M, g k ) remains bounded and the associated volume measures will not develop a Dirac mass. Using the sequence of metrics (g k = e -2f k g 0 ) k∈N , we get an analogous example where the metric vanishes at x 0 as k → +∞.

Analogously, one can construct examples of sequences of conformal deformations whose volumes and L n 2 norms of the scalar curvatures are uniformly bounded, for which the limit metric blows up (or goes to 0) at any countable (even dense) set of points.

The reader may easily show that in this example, eventhough the sequence of conformal factors is unbounded, the sequence of associated distances (d g k ) k∈N converges for the C 0uniform topology to the Riemannian distance associated to the (singular) metric g ∞ = e 2f∞ g 0 on M .

Our first theorem is a precompactness result which follows from an elementary application of Sobolev embeddings.

Theorem 1.1. Let (M, g 0 ) be a compact smooth Riemannian manifold and let M δ V be the space of conformal smooth metrics on M , g f = e 2f g 0 with associated Riemannian measure dµ f = e nf dvol g0 , such that there exists V > 0, δ > 0 such that:

(1) µ f (M ) ≤ V , and [START_REF] Aubin | Problèmes isopérimétriques et espace de Sobolev[END_REF]

(Scal g f ) + L n 2 (M,dµ f ) ≤ α(n, 2) -δ.
Then for some α ∈ (0, 1), the set of distances {d g f ; g f ∈ M δ V } is precompact for the C α topology on M × M . In particular, the set of metric spaces {(M, d g f ), g f ∈ M δ V } is precompact for the Gromov-Hausdorff topology.

It follows from the Example 1 presented above that the upper bound

(Scal g f ) + L n 2 (M,dµ f ) ≤ α(n, 2) -δ = σ 2 n
n n(n -1)δ is optimal to get C α precompactness of the sequence of distances.

Theorem 1.1 does not exclude the possible collapsing of some part of the manifold in the limit of such conformal deformations. We will show in Corollary 2.4 that adding to the previous hypotheses a lower bound on the volume and a L p bound on the scalar curvature for some p > n 2 is enough to recover C α precompactness of the sequence of Riemannian metrics as in [START_REF] Gursky | Compactness of conformal metrics with integral bounds on curvature[END_REF].

The rest of our paper is devoted to the exploration of conformal deformations for which only volume bounds and L n 2 bounds on the scalar curvature are imposed. We will see in particular that, no collapsing can occur if the L n/2 norm of the scalar curvature is small at small scales.

Our ideas come from a paper by Yi Wang [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF], who was interested in conformal deformations with integral bounds on the Q-curvature related to the so-called Muckenhoupt weights, or A ∞ -weights. If (M, g) is a closed manifold, a non-negative L 1 loc function w is called an A ∞ -weight with respect to g if there exist C > 0, q > 1 such that for all g-geodesic balls B ⊂ M , B w q dµ g 1 q ≤ C B wdµ g , where µ g is the volume measure associated to g. Several equivalent characterizations of these A ∞ -weights will be presented in Section 3.

Theorem 1.2 below establishes that, under L n/2 pinching conditions on the scalar curvature closely related to those in Theorem 1.1, the volume densities e nf are uniformly A ∞ weights with respect to g 0 . Theorem 1.2. Let (M n , g 0 ) be a closed Riemannian manifold, R 0 ∈ (0, diam(M, g 0 )] and δ, Λ > 0. Let f : M → R be a smooth function such that the metric g f = e 2f g 0 and its associated Riemannian measure

dµ f = e nf dvol g0 satisfy i) ´M Scal g f n 2 dµ f ≤ Λ. ii) For any x ∈ M , ˆB(x,R0) Scal g f n 2 + dµ f 2 n ≤ α(n, 2) -δ.
Then e nf is an A ∞ -weight with respect to g 0 , with constants only depending on n, g 0 , R 0 , δ and Λ.

The hypotheses as well as the conclusion of Theorem 1.2 are scale invariant. We will see in Section 3 that if we fix upper and lower volume bounds, having A ∞ control for e nf implies C α bounds for the distance associated to the metric e 2f g, for some α > 1. This implies as well uniform control on certain analytic quantities associated to the conformal metric. Denoting g f = e 2f g 0 , the fact that e nf is an A ∞ weight implies that for any g 0geodesic ball, its g f -diameter is bounded from above in terms of its g f -volume and that the measure dµ f = e nf dµ g0 is doubling, with constants only depending on δ, Λ and R 0 .

Notice that in the second family of examples presented above, for which Theorem 1.2 applies, the Riemannian volumes of the members of the family satisfy a uniform Strong A ∞ bound in the sense of [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF], see Definition 4.1 in Section 4 below. This implies that for a sequence of metrics coming from these examples, the Gromov-Hausdorff limit (M ∞ , d ∞ ) is homeomorphic to M and endows M with a distance d ∞ that is uniformly bi-Hölder to (M, d g0 ). In addition, the metrics in this family of examples have uniform Sobolev, Poincaré and isoperimetric inequalities. Our main result shows that this is always the case, provided the L n/2 -pinching of the scalar curvature is small enough at small scales.

Theorem 1.3. Let (M n , g 0 ) be a closed Riemannian manifold. Then there exists a constant Λ 0 = Λ 0 (g 0 ) with the following property: Let g f be given by e 2f g 0 with f ∈ C 2 (M ). If there is a R 0 ∈ (0, diam(M, g 0 )] such that

(2) ∀x ∈ M, ˆB(x,R0) Scal g f n 2 dµ f ≤ Λ 0
then e nf is a strong A ∞ weight with respect to g 0 , with constants only depending on g 0 , R 0 and Λ 0 . Moreover, for all α ∈ (0, 1), the distances d f and d 0 are uniformly α-bi-Hölder with constants depending only on g 0 , R 0 , Λ 0 and α.

Before we continue, let us introduce some notation. Let (M n , g 0 ) be a closed Riemannian manifold, for v, V, δ, Λ > 0 and R 0 ∈ (0, diam(M, g 0 )], define M Λ v,V,R0 as the set of smooth functions f on M such that the associated metric

g f = e 2f g 0 satisfies i) v ≤ Vol(M, g f ) ≤ V ii) for any x ∈ M : ´B(x,R0) Scal g f n 2 dµ f 2 n ≤ Λ.
As mentioned previously, strong A ∞ -weight control coupled with volume bounds has many interesting geometric consequences. Some of them are listed in the following corollary.

Corollary 1.4. Let (M n , g 0 ) be a closed Riemannian manifold. Then there exists Λ 0 = Λ(g 0 ) such that for all R 0 ∈ (0, diam(M, g 0 )] and any v, V with 0 < v ≤ V , the set of metric spaces

{(M, d g f ) ; f ∈ M Λ0 v,V,R0
} is precompact for the Gromov-Hausdorff topology. In addition, any sequence in this set has a subsequence whose limit (M, d ∞ ) is α-bi-Hölder to (M, d g0 ) for all α ∈ (0, 1).

It is worth noticing that the conclusion of Theorem 1.2 is weaker than the conclusion of Corollary 1.4 but holds under an explicit (and optimal, due to examples of [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF]) L n/2pinching of the scalar curvature. The constant Λ(g 0 , R 0 ) can be explicited in terms of the Ricci curvature of g 0 and its diameter. We could not produce a sequence of examples satisfying the hypotheses of Theorem 1.2 which would not be uniformly Strong A ∞ .

Remark 1.5. Our proof of Theorem 1.3 shows that, under the same hypotheses, e nf is actually a stronger A ∞ weight in the sense of [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF]. This implies by Theorem 5.2 of [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF] that there exists N > 0 such that any such (M, g f = e 2f g 0 ) can be embedded in R N via a bi-Lipschitz embedding, with uniform bi-Lipschitz constant. We will not further discuss this fact here.

Eventhough our initial objective was to study conformal deformations of closed manifolds, our techniques also provide an interesting criterion to get strong A ∞ on (R n , eucl).

Theorem 1.6. Let g = e 2f eucl be a conformal deformation of the Euclidean metric on R n such that:

• Vol(R n , g) = +∞, • ˆRn | Scal g | n/2 dµ g < +∞.
Then e nf is a strong A ∞ weight on R n with respect to eucl.

As mentionned above, being a strong A ∞ weight has many implications in terms of geometric control. In particular, this theorem has the following striking geometric corollary, which seems to have remain unknown.

Corollary 1.7. Let g = e 2f eucl be a conformal deformation of the Euclidean metric on R n such that:

• Vol(R n , g) = +∞,

• ˆRn | Scal g | n/2 dµ g < +∞.
Then there are positive constants θ, γ such that any g-geodesic ball B g (x, r) satisfies

θ -1 r n ≤ µ g (B g (x, r)) ≤ θr n .
Moreover, (R n , g) satisfies the Euclidean isoperimetric inequality: for any bounded smooth open subset Ω ⊂ R n we have:

γ µ g (Ω) 1-1 n ≤ µ g (∂Ω).
Let us conclude by an intricate question. Let (g k = e 2f k g 0 ) k∈N be a sequence of conformal deformations of g 0 with

f k ∈ M Λ0 v,V,R0
, where Λ 0 satisfies the conclusion of Corollary 1.4. Then, up to extracting a subsequence, the sequence of associated metric spaces (M, d g k ) converges in Gromov-Hausdorff topology to some distance d ∞ on M , which is bi-Hölder equivalent to d 0 . Moreover, a subsequence of the Riemannian volume measures µ k = dµ g k = e nf k dµ g0 converges in the weak- * topology to a positive measure µ ∞ on M . Since all the µ k are uniformly strong A ∞ -weights with respects to dvol g0 , so is µ ∞ : there exists e nf∞ ∈ L 1 (M ) such that µ ∞ = e nf∞ dvol g0 and µ ∞ is still a strong A ∞ -weight with respect to dvol g0 . Hence the weight e nf∞ induces another distance d f∞ on M , analogous to a conformal Riemannian distance (see Section 7 below or [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF] for a precise definition).

A natural question is: Under which assumption do the distances d ∞ and d f∞ coincide? We could not give a full answer to this question. We know that they do coincide in Example 2 presented above. On the other hand, in the last section of this paper we will describe another example which shows that uniform strong A ∞ -weight control is not enough to establish this equality in general. Nevertheless, volume bounds and a L n/2 bound on the scalar curvature give a much stronger control on the geometry than just strong A ∞ -weight controls. Thus, the distances could still coincide under the integral pinching of the scalar curvature under consideration.

From now on, given a background Riemannian manifold (M, g 0 ), for any conformal deformation g f = e 2f g 0 , we will denote by d f = d g f the associated Riemannian distance and by dµ f = dvol g f the associated Riemannian volume measure. In particular, d 0 and µ 0 are the distances and volume measure associated to g 0 , respectively.

Acknowledgements: The authors are grateful for the support of the ANR grants ACG (ANR-10-BLAN 0105) and CCEM (ANR-17-CE40-0034) which made this collaboration possible. C.L. Aldana was also supported by the Fonds National de la Recherche, Luxembourg 7926179.

INTEGRAL BOUNDS FOR THE SCALAR CURVATURE AND CONVERGENCE OF

DISTANCES

This first section is devoted to the proof of Theorem 1.1. This theorem is an immediate consequence of Propositions 2.2 and 2.3 given below. The proof will show that for any sequence of metrics (g n ) n∈N in M δ V , the sequence of distances (d gn ) n∈N converges in some Hölder topology up to extracting a subsequence. This implies the precompactness of the sequence of metric spaces (M, d gn ) n∈N for the Gromov-Hausdorff topology, as shown by the following lemma which will be used several times in the sequel. Lemma 2.1. Let (X, d 0 ) be a compact metric space, α ∈ (0, 1), C > 0, and (d n ) n∈N be a sequence of distances on M which is bounded in the α-Hölder topology with respect to d 0 , i.e. such that for all x, x ′ , y, y ′ ∈ M and all n ∈ N,

|d n (x, y) -d n (x ′ , y ′ )| ≤ C (d 0 (x, x ′ ) α + d 0 (y, y ′ ) α ) .
Then up to extracting a subsequence, d n is converging uniformly and in the α ′ -Hölder topology for all α ′ ∈ (0, α) to a map d ∞ : M × M → [0, +∞). Moreover, the sequence of metric spaces (M, d n ) converges in the Gromov-Hausdorff topology to the metric space

(M ∞ , d ∞ ) where M ∞ = M/ ∼, with x ∼ y if and only if d ∞ (x, y) = 0.
Proof. Let (d n ) n∈N be a sequence of distances on M satisfying the above hypotheses. By Arzela-Ascoli Theorem, it has a subsequence which converges in the C 0 -uniform topology on M × M to some non-negative map d ∞ : M × M → [0, +∞), satisfying the triangular inequality. By construction, d ∞ induces a distance on M ∞ = M/ ∼, with x ∼ y if and only if d ∞ (x, y) = 0. The C 0 -uniform convergence of the d k implies that the sequence of metric spaces (M, d n ) converges in the Gromov-Hausdorff topology to the metric space (M ∞ , d ∞ ), see for instance [START_REF] Burago | A course in metric geometry[END_REF] p.260, Example 7.4.4.

An elementary computation shows that the convergence of (d n ) is actually uniform in all C α ′ Hölder spaces for all 0 < α ′ < α, which concludes the proof of our lemma. Proposition 2.2. Let (M, g 0 ) be a compact Riemannian manifold and let (g k = e 2f k g 0 ) k≥0 be a sequence of smooth conformal metrics. Assume that there exist p > n and A > 0 such that for all k ∈ N, ˆM e pf k dµ 0 ≤ A.

Then the sequence of distances (d f k : M × M → [0, +∞)) k∈N is precompact in the α-Hölder topology for all α ∈ (0, 1 -n p ). In particular, the sequence of metric spaces (M, d f k ) k∈N is precompact for the Gromov-Hausdorff topology.

Proof. Let x ∈ M be fixed. Recall that for any smooth Riemannian metric g on M , ||∇ g d g (x, .)|| g = 1 almost everywhere, see e.g. [START_REF] Sakai | Riemannian Geometry[END_REF]Prop. 4.8]. Let g f = e 2f g 0 be conformal metric, we have hence for almost all y = x,

||∇ g0 d f (x, .)|| g0 (y) = e f (y) .
Therefore, there exists C > 0 such that

(3) ˆM e pf dµ 0 ≤ A ⇒ ||d f (x, .)|| W 1,p (M,g0) ≤ CA,
where W 1,p (M, g 0 ) is the Sobolev space of functions on M with distributional derivative in L p (g 0 ). Now, let (g k = e 2f k g 0 ) k≥0 be a sequence of smooth conformal metrics. Assume that there exist p > 1 and A > 0 such that for all k ∈ N, ˆM e pf k dµ 0 ≤ A.

It follows from (3) that the sequence of distances (d f k (x, .)) k∈N is bounded in W 1,p (M, g 0 ). Therefore, since for all k ∈ N, d f k (x, .) vanishes at x, by Sobolev embeddings the sequence is bounded in C α with α = 1 -n p by some constant C ′ independent of x. We have hence, for all x, x ′ , y, y ′ ∈ M and all n ∈ N,

|d f k (x, y) -d f k (x ′ , y ′ )| = |d f k (x, y) -d f k (x ′ , y) + d f k (x ′ , y) -d f k (x ′ , y ′ )| ≤ C ′ (d 0 (x, x ′ ) α + d 0 (y, y ′ ) α ) .
By Lemma 2.1, this implies Proposition 2.2.

We now show that any conformal deformation of g 0 whose scalar curvature has a positive part with L n/2 norm strictly less than the round sphere has a conformal factor that satisfies the integrability condition required in Proposition 2.2. Proposition 2.3. Let (M n , g 0 ) be a closed Riemannian manifold. Let V, δ be positive real numbers. There exists

A = A(V, δ) > 0 such that for all ε > 0 small enough, if i) Vol(M, g f ) ≤ V, ii) (Scal g f ) + L n 2 (M,dµ f ) ≤ α(n, 2) -δ then (4)
ˆM e n(1+ǫ)f dµ g0 ≤ A.

Proof. We are going to use the following optimal Sobolev inequality du to E. Hebey and M. Vaugon ( [START_REF] Hebey | Meilleures constantes dans le théorème d'inclusion de Sobolev[END_REF]) (this inequality had been conjectured by T. Aubin [START_REF] Aubin | Problèmes isopérimétriques et espace de Sobolev[END_REF]): if we write again α(n, 2) = σ

2 n
n n(n -1), then there is a constant B such that for all ϕ ∈ C ∞ (M ),

(5) α(n, 2) ˆM ϕ 2n n-2 dµ 0 1-2 n ≤ 4(n -1) n -2 ˆM |dϕ| 2 g0 dµ g0 + ˆM Bϕ 2 dv g0 . If g f = u 4 n-2 g 0 if follows from [43] that (6) Scal g f u 4 n-2 u = 4(n -1) n -2 ∆ g0 u + Scal g0 u.
In particular for any ǫ > 0 we get

4(n -1) n -2 ∆ g0 u 1+ǫ + (1 + ǫ) Scal g0 u 1+ǫ + (n -1) 4ǫ(1 + ǫ) n -2 u ǫ-1 |du| 2 g0 = (1 + ǫ) Scal g f u 4 n-2 u (1+ǫ) .
Using the Hebey and Vaugon Sobolev inequality for ϕ = u (1+ǫ) , we get

α(n, 2) ˆM u (1+ǫ) 2n n-2 dµ g0 1-2 n ≤ 4(n -1) n -2 ˆM u (1+ǫ) ∆ g0 u 1+ǫ dµ g0 + B ˆM u 2(1+ǫ) dµ g0 ≤ (1 + ǫ) ˆM (Scal g f ) + u 4 n-2 u 2(1+ǫ) dµ g0 +((1 + ǫ)S 0 + B) ˆM u 2(1+ǫ) dµ g0 ,
where S 0 = Scal g0 L ∞ . When ǫ ≤ 2/(n -2), using the Hölder inequality we get

(α(n, 2) -(1 + ǫ) (Scal g f )+ L n 2 (M,dµ f ) ˆM u (1+ǫ) 2n n-2 dµ g0 1-2 n ≤ ((1 + ǫ)S 0 + B) (Vol g0 (M )) 2-(n-2)ǫ n Vol g f (M ) (1+ǫ) n-2 n , which gives with e 2f = u 4 n-2 , ˆM e n(1+ǫ)f dµ g0 ≤ ((1 + ǫ)S 0 + B) (Vol g0 (M )) 2-(n-2)ǫ n δ -ǫ(α(n, 2) -δ) V (1+ǫ) n-2 2 ,
which is precisely (4) when ǫ is small enough, say ǫ < δ α(n,2)-δ .

Notice that under our very weak hypotheses, the limit space (M ∞ , d ∞ ) obtained as the Gromov-Hausdorff limit of the sequence (M, d g k ) k∈N may be very singular; the singularities depending on the set where d ∞ vanishes. Even if the volume of (M, g k ) remains constant (recall that the L n/2 bound of the scalar curvature is invariant under rescaling), we cannot a priori prevent collapsing of some open part of the manifold. In the sequel, we look for stronger conditions on the scalar curvatures and the volumes so that we get a better geometric picture of the limit.

Let us first note that if we add to the hypotheses in Theorem 1.1 a lower bound on the volume and a uniform L p bound on the scalar curvature (and not the full Riemann tensor) with p > n 2 , then we get C α -precompactness for the conformal factors:

Corollary 2.4. Let (M n , g 0 ) be a closed Riemannian manifold. Let v, V, Λ, δ be positive real numbers and let p > n/2. Then the set

M p,Λ,δ v,V of smooth functions f ∈ C ∞ (M ) such that g f = e 2f g 0 satisfies i) v ≤ Vol(M, g f ) ≤ V, ii) ´M Scal g f p dµ f ≤ Λ iii) (Scal g f ) + L n 2 (M,dµ f ) ≤ σ 2 n n n(n -2) -δ is precompact in C α (M ) for 0 < α < 2p-n p .
Proof. It follows from Proposition 2.3 that for all metrics g f = e 2f g 0 in M p,Λ,δ v,V , the volume density e nf satisfies a L 1+ε uniform bound given by [START_REF] Besse | Einstein manifolds[END_REF].

Combining this uniform stronger integrability of e f together with L p bounds for the scalar curvature and volume bounds, a Harnack inequality due to Trudinger in [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] together with elliptic regularity will give C α -precompactness for the space of metrics for 0 < α < 2p-n p . This is explained with details in Section 2.2 of [START_REF] Matthiesen | Regularity of conformal metrics with large first eigenvalue[END_REF]. In particular, any sequence of metrics satisfying some L p bound on the scalar curvature, p > n/2, and the above L n/2 bounds on its positive part, converges (up to extraction of a subsequence) in some C α -topology to a Riemannian metric g ∞ = e 2f∞ g 0 , where f ∞ ∈ C α (M ). This slightly improves the well known result by M. Gursky given in [START_REF] Gursky | Compactness of conformal metrics with integral bounds on curvature[END_REF].

The rest of our paper is devoted to the study of sequences of conformal deformations with fixed volume bounds and a mere L n/2 bound on the scalar curvature. It follows from [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF] that in this setting, one cannot get C 0 precompactness for the sequence of conformal factors (or, equivalently, for the sequence of Riemannian metrics). Nevertheless, we will show that the sequence of associated measured metric spaces and the limit metric spaces (M ∞ , d ∞ ) thus obtained satisfy many uniform geometrical properties.

OPTIMAL L n/2 BOUNDS FOR THE SCALAR CURVATURE AND A ∞ WEIGHTS

In this section, we study sequences (g k = e 2f k g 0 ) k∈N of conformal metrics with a uniform upper bound on the volumes, such that the L n/2 -norms of the positive part of the scalar curvature on small spheres is uniformly bounded by the one of the standard sphere, and which moreover have some uniform L n/2 bound for the total scalar curvature. We will show that the Riemannian volume densities {e nf k } k∈N are then uniformly A ∞ -weights with respect to g 0 , see Definition 3.1.

To be uniformly A ∞ -weights has several nice geometrical consequences. First of all, the sequence of volume measures (dµ f k ) k∈N is uniformly doubling with respect to the initial distance d 0 . In particular, this rules out the kind of blowups presented in Example 1 in our introduction. More precisely, we will see that the A ∞ control implies that the Riemannian distances d f k are uniformly controlled in terms of the volumes dµ f k , locally. We will show as well Gromov-Hausdorff precompactness for the set of conformal metrics satisfying a uniform A ∞ -weight bound, independently of any control on the curvature.

In the afore mentioned work of Gursky [START_REF] Gursky | Compactness of conformal metrics with integral bounds on curvature[END_REF], a key step to get C α compactness for a sequence of conformal metrics (g k = e 2f k g 0 ) k∈N whose Riemann curvatures satisfy a uniform L p bound (p > n/2) is to show that if g 0 is not conformal to the standard sphere, then the conformal factors in such sequences satisfy a uniform Harnack inequality: there exists C > 1 such that for all k ∈ N,

sup e f k inf e f k ≤ C.
Uniform volume bounds and (even very small) L n/2 bounds on the scalar curvature cannot ensure such Harnack inequality, as shows Example 2 in our introduction. A uniform A ∞ control on the weight e nf k is actually a local L q integral version of this Harnack inequality for some q > 1.

3.1.

A ∞ weights and conformal metrics. We now give a presentation of A ∞ weights on a closed Riemannian manifold. The reader can find a good exposition and more details in [36, chapter V]. Then, we will draw some geometric consequences for conformal metrics having such A ∞ control.

Let (M n , g 0 ) be a closed Riemannian manifold. Throughout this document we will denote by B(x, r) : the g 0 -geodesic ball centered at x and of radius r.

We will sometimes write B for a g 0 -geodesic ball whose radius will be denoted by r(B).

For θ > 0, θB will be the ball with the same center and with radius θr(B). D will denote the diameter of (M n , g 0 ). The measure µ 0 will still be the g 0 Riemannian volume. If E ⊂ M is a measurable set and f ∈ L 1 (E, dµ 0 ) the average of f over E will be denoted by

E f dµ 0 = 1 µ 0 (E) ˆE f dµ 0 . Definition 3.1. We say that a non negative function w ∈ L 1 (M, dµ 0 ) is a A ∞ weight with respect to g 0 if

one of the following equivalent properties is satisfied:

i) There is a q > 1 and a constant C such that for any geodesic ball B ⊂ M , the following reverse Hölder inequality with exponent q holds:

B w q dµ 0 1 q ≤ C B wdµ 0 .
ii) There is p > 1 and a constant C such that for any geodesic ball B ⊂ M :

B wdµ 0 B w -1 p-1 dµ 0 p-1 ≤ C.
If w satisfies this condition, it is called A p -weight. iii) There are constant δ, ε ∈ (0, 1) such that for any geodesic ball B ⊂ M and any

E ⊂ M if E ⊂ B satisfies µ 0 (E) ≤ δµ 0 (B) then ´E wdµ 0 ≤ ε ´B wdµ 0 . iv)
There are constants α > 1 and C > 0 such that for any geodesic ball B ⊂ M and any E ⊂ B:

1 C µ 0 (E) µ 0 (B) α ≤ ´E wdµ 0 ´B wdµ 0 ≤ C µ 0 (E) µ 0 (B) 1 α .
Remark 3.2. The different constants in the different definitions are mutually controlled.

In particular, we will use the following fact: If w is an A p weight satisfying part ii) above for all p > 1, then it fulfills reverse Hölder inequality i) with exponent q for all q > 1, with mutually controlled constants. See [36, p.196-203] for a proof.

It is possible to show that the properties in Definition 3.1 hold for all geodesic balls if and only if they hold for geodesic balls of radius less than some fixed R 0 > 0. We will prove this statement for the Reverse Hölder inequality. In order to do that, we need the following general result. d,µ) is a metric measure space whose balls B with radius less than R 0 satisfy the doubling condition:

Lemma 3.3. If (X,
µ(2B) ≤ θµ(B)
then for any R 1 ≥ R 0 there is a constant θ ′ depending only on θ and R 1 /R 0 such that any ball B with radius less than R 1 satisfies the doubling condition:

µ(2B) ≤ θ ′ µ(B) Proof. Indeed let B(x, R) a ball of radius R ∈ [R 0 , R 1 ], then from [11, Lemma 3.10], we have µ(B(x, 2R)) ≤ θ 50+50 2R R 0 µ f (B(x, R 0 )) ≤ θ 50+50 2R 1 R 0 µ(B(x, R)).
Lemma 3.4. Let q > 1, and C, R 0 > 0 be fixed. Let w be a uniformly (q, R 0 , C)-A ∞ weight, i.e. every geodesic ball B of radius r ≤ R 0 satifies:

B w q dµ 0 1 q ≤ C B wdµ 0 .
Then the measure wdµ 0 has the doubling volume property and there exists C ′ = C ′ (q, C, R 0 , g 0 ) > 0 such that for all balls B of radius r ≥ 0, one has

(7) B w q dµ 0 1 q ≤ C ′ B wdµ 0 .
Proof. Assume that w satisfies the hypotheses of the lemma for some q > 1 and C, R 0 > 0. We first show that wdµ 0 has the doubling volume property. Let us first consider geodesic balls of radius less than R 0 > 0. We start by showing that there is θ > 0 such that for the measure dω = wdµ we have the doubling property:

r(B) ≤ R 0 /2 =⇒ ω(2B) ≤ θω(B).
Indeed using both Hölder and Reverse Hölder inequalities we easily get for any τ ∈ (0, 1) :

ω(B \ τ B) ω(B) ≤ C µ 0 (B \ τ B) µ 0 (B) 1-1 q .
The manifold M is compact hence there is a constant B such that for any geodesic ball :

µ 0 (B \ τ B) ≤ B(1 -τ )µ 0 (B).
We choose τ ∈ (0, 1) such that C (B(1τ ))

1-1 q = 1 2 and we get ω(B) ≤ 2ω(τ B), therefore ω(B) ≤ 2 N ω(τ N B). Choosing now N such that τ N ≤ 1 2 < τ N -1 gives the result with θ = 2 N .
As the diameter of (M, g 0 ) is finite, we deduce from the Lemma 3.3 that the measure ω is doubling: there is a constant θ such that for any ball B ⊂ M :

ω(2B) ≤ θω(B).
We can now show that the reverse Hölder inequality holds for any ball. Let B be a geodesic ball of radius r ∈ (R 0 , D], then we can find a minimal family of geodesic balls B α = B(x α , R 0 /2) of radius R 0 /2 such that B ⊂ ∪ α B α , and the balls B(x α , R 0 /4) are disjoint, hence include in 2B. Since w satisfies reverse Hölder on the balls B(x α , R 0 /2), we get

ˆBα w q dµ 0 1 q ≤ Cµ 0 (B α ) 1 q µ 0 (B α ) ˆBα wdµ 0 = Cµ 0 (B α ) 1 q -1 ˆBα wdµ 0 .
From the doubling condition on (M, d 0 , µ 0 ) there is a constant such that

µ 0 (B) ≤ Cµ 0 (B α ).
Hence

ˆBα w q dµ 0 1 q ≤ Cµ 0 (B) 1 q -1 ˆBα wdµ 0 ˆB w q dµ 0 ≤ α ˆBα w q dµ 0 ≤ C α µ 0 (B α ) 1-q ˆBα wdµ 0 q ≤ Cµ 0 (B) 1-q α ˆBα wdµ 0 q ≤ Cµ 0 (B) 1-q θ q α ω( 1 2 B α ) q ≤ Cµ 0 (B) 1-q θ q α ω( 1 2 B α ) q ≤ Cµ 0 (B) 1-q θ q ω(2B) q ≤ Cµ 0 (B) 1-q θ 2q ω(B) q ,
from which the conclusion follows.

Let now f ∈ C ∞ (M ) and g f = e 2f g 0 be a Riemannian metric conformal to g 0 . A key geometric consequence of having A ∞ control on the weight e nf is given in the following lemma.

Lemma 3.5. Let f : M → R be a smooth map. Assume that w = e nf is an A ∞ weight satisfying the above reverse Hölder inequality with constants C > 0 and q > 1. Then there is a constant B, depending only on C, q and (M, g 0 ), such that for any x, y ∈ M :

d f (x, y) n ≤ B ˆB(x,d0(x,y))
e nf dµ 0 .

This lemma has been shown on the Euclidean space by G. David and S. Semmes, see [17, inequality (1.2)] and [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF]Proposition 3.12b].

Proof. By Sobolev inequality (see for instance [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] p.148), there is a constant C > 0 depending only on the geometry of (M, g 0 ) and q, such that for p = qn, we have for any balls B ⊂ M and any ϕ ∈ W 1,p (2B) and x, y ∈ B:

|ϕ(x) -ϕ(y)| ≤ Cr(B) 1-n p ˆ2B |dϕ| p dµ 0 1 p .
If we use this for the function ϕ = d f (x, .) and B a ball centered at some point m with d 0 (x, m) = d 0 (m, y) = d 0 (x, y)/2 and with radius 3 4 d 0 (x, y). we will get

d f (x, y) ≤ Cr(B) 1-n p ˆ2B e qnf dµ 0 1 qn
Then the conclusion follows from the reverse Hölder inequality and the fact that the µ 0 measure of 2B is comparable to r(B) n .

Let us show now that the space of conformal metrics with uniform A ∞ bounds is precompact in the Gromov-Hausdorff topology. We will not need this result in the sequel since Theorem 1.1 will already ensure Gromov-Hausdorff precompactness once we fix a suitable L n/2 bound on the scalar curvature. Nevertheless, this is an easy consequence of the measure being A ∞ weights which seems to have remained unnoticed. Proposition 3.6. Let (M n , g 0 ) be a closed Riemannian manifold let C, V > 0, R 0 ∈ (0, diam(M, g 0 )] and q > 1.

Let M V,R0,q,C be the set of smooth functions such that f ∈ M V,R0,q,C satisfies the following conditions: i) for the Riemanian metric

g f = e 2f g 0 , Vol(M, g f ) ≤ V ; ii) w = e nf is (R 0 , q, C)-uniformly A ∞ i.e.
for any geodesic ball B of radius less or equal than R 0 we have:

B w q dµ 0 1 q ≤ C B wdµ 0 .
Then the set of distances {d f , f ∈ M V,R0,q,C } is pre-compact in the C α topology for all α ∈ (0, 1 -1 q ). In particular, the set of metric spaces {(M, d f ), f ∈ M V,R0,q,C } is precompact for the Gromov-Hausdorff topology.

Proof. By Lemma 3.4, there exists C ′ > 0 such that for all f ∈ M v,V,R0,q,C and all geodesic balls B (of any radius),

B w q dµ 0 1 q ≤ C B wdµ 0 .
In particular, if the radius of B is larger than the diameter of (M, g 0 ), we get that for all

f ∈ M v,V,R0,q,C , ˆM e nqf ≤ C ′ (Vol(M, g 0 )) 1 q -1 V.
The conclusion follows then immediately from Proposition 2.2.

3.2.

A ∞ control from L n/2 bound on the scalar curvature. We now show that a L n/2 bound on the scalar curvature, with L n/2 bound on the positive part of the scalar curvature uniformly smaller than what it is for the standard sphere at some (even small) fixed scale, implies a uniform A ∞ control of the conformal factor. Once again, it follows from the Example 1 presented in our introduction that these integral bounds are optimal to get such A ∞ control. Theorem 3.7 below implies Theorem 1.2 of the introduction and explicit its result.

Theorem 3.7. Let (M n , g 0 ) be a closed Riemannian manifold, δ, Λ > 0 and R 0 > 0.

Then there exist p = p(n, g 0 , R 0 , Λ) > 0 and C = C(n, g 0 , R 0 , δ, Λ) > 0 such that for any smooth function f : M → R whose associate metric

g f = e 2f g 0 satisfies i) ´M Scal g f n 2 dµ f ≤ Λ, and
ii) for any x ∈ M :

ˆB(x,R0) Scal g f n 2 + dµ f 2 n ≤ α(n, 2) -δ,
we have that for any g 0 -geodesic ball B of radius less that R 0 /2,

ˆB e nf dµ 0 ˆB e -nf p dµ 0 p ≤ C.
In particular, e nf is an A ∞ -weight with respect to g 0 , with constants only depending on n, g 0 , R 0 , δ and Λ.

Proof. The proof of this theorem was directly inspired by the proof of the Harnack inequality for positive solution of second order elliptic equation, see [20, theorem 8.20] and its proof. Let R 0 ∈ (0, diam(M, g 0 ]), δ, Λ > 0 and let f : M → R be a smooth function satisfying the hypotheses above. We write again g f = u

= ε(n, g 0 , R 0 , Λ) > 0 and C = C 1 (n, g 0 , R 0 , Λ) > 0 such that for any g 0 -geodesic ball B of radius less than R 0 /2, B u ε dµ 0 B u -ε dµ 0 ≤ C.
Proof. Since we consider g f = u 4 n-2 g 0 , let us recall equation 6:

Scal g f u 4 n-2 u = 4(n -1) n -2 ∆ g0 u + Scal g0 u Set V = Scal g f u 4 n-2 , W = V -Scal g0 and λ 0 = Scal g0 L ∞ . We have 4(n -1) n -2 ∆ 0 ln(u) = W + 4(n -1) n -2 |d ln(u)| 2 g0 .
Now let B and ξ be a Lipschitz cut-off function with support in 2B such that ξ = 1 on B and |dξ| ≤ r(B) -1 . Then

ˆM ξ 2 W + 4(n -1) n -2 |d ln(u)| 2 g0 dµ 0 = 4(n -1) n -2 ˆM 2ξ dξ, d ln(u) dµ 0 ≤ 4 ˆM ξ 2 |d ln(u)| 2 g0 dµ 0 + 16(n -1) 2 (n -2) 2 ˆM |dξ| 2 dµ 0 By Hölder inequality, we have ˆM ξ 2 (-V )dµ 0 ≤ Scal - g f L n/2 (2B) µ 0 (2B) n-2 2 ,
Moreover, for g 0 there is an uniform constant ω such that for any r ∈ (0, diam(M, g 0 )] and any x ∈ M :

(8) 1 ω r n ≤ µ 0 (B(x, r)) ≤ ωr n .
Hence we get

4(n -1) n -2 -4 ˆB |d ln(u)| 2 g0 dµ 0 ≤ Cλ 0 r(B) n +C 1 + Scal - g f L n 2 (2B) µ 0 (2B)r(B) -2 .
In particular we have

(9) r(B) 2 B |d ln(u)| 2 g0 dµ 0 ≤ C 1 + Scal - g f L n 2 (2B)
.

There is a r 0 > 0 such that all the geodesic balls B(x, r 0 ) are almost Euclidean: i.e. there is a smooth map ϕ x : B(0, r 0 ) → B(x, r 0 ) with

1 2 eucl ≤ ϕ * x g 0 ≤ 2eucl,
where eucl is the Euclidean metric and B(0, r 0 ) the Euclidean ball of radius r 0 . Hence there are positive constants θ and λ that only depends on n such that if B is a g 0 geodesic ball of radius r(B) ≤ r 0 /2 then

µ 0 (2B) ≤ θµ 0 (B)
and

∀φ ∈ C 1 (B) : ˆB φ - B φ 2 dµ 0 ≤ λr(B) 2 ˆB |dφ| 2 dµ 0 .
We can assume that R 0 ≤ r 0 , hence the above estimation (9) and the above Poincaré inequality implies a BMO (bounded mean oscillation) estimate on the function ln(u). Moreover, we also get from ( 9) that for all balls B with radius R ≤ R 0 ,

ˆB |d ln(u)| g0 dµ 0 ≤ CR n/2 ˆB |d ln(u)| 2 g0 dµ 0 1/2 ≤ CR n-1 .

Therefore by

Therefore

B u ε dµ 0 B u -ε dµ 0 ≤ Ce ε(ln u) B e -ε(ln u) B ≤ C.
So far we have only used an estimate of the negative part of the scalar curvature of g f in L n/2 . However, we will need the L n/2 bound on the positive part of the scalar curvature in order to get the following estimate. Lemma 3.9. Assume the same hypothesis of Theorem 3.7. For all ε > 0, there exists C = C(n, g 0 , R 0 , δ, Λ, ε) > 0 such that for all g 0 -geodesic balls of radius less than R 0 ,

1 2 B u 2n n-2 dµ 0 ≤ C B u ε dµ 0 2n ε(n-2)
.

Proof. The proof of this lemma is done in two stages, the first one being a localization of what we did in the proof of Proposition 2.3. Let R, r ≥ 0 such that r + R ≤ R 0 and let ξ be a cut-off function with support in B(x, r + R) such that

• 0 ≤ ξ ≤ 1, • |dξ| g0 ≤ 1 r • ξ = 1 on B(x, R)
We have the equality

4(n -1) n -2 ˆB(x,R+r) |d(ξu)| 2 g0 dµ 0 = 4(n -1) n -2 ˆB(x,R+r) (∆ g0 u) uξ 2 dµ 0 + 4(n -1) n -2 ˆB(x,R+r) |dξ| 2 g0 u 2 dµ 0 = ˆB(x,R+r) V u 2 ξ 2 dµ 0 - ˆB(x,R+r) Scal g0 u 2 ξ 2 dµ 0 + 4(n -1) n -2 ˆB(x,R+r) |dξ| 2 g0 u 2 dµ 0 .
We use now the Hebey-Vaugon Sobolev inequality (5) and we get that

α(n, 2) ˆB(x,R+r) (ξu) 2n n-2 dµ 0 1-2 n ≤ 4(n -1) n -2 ˆB(x,R+r) |d(ξu)| 2 dµ 0 + B ˆB(x,R+r) (ξu) 2 dµ 0 (10)
Hence with Hölder inequality, we get α(n, 2)

ˆB(x,R+r) (ξu) 2n n-2 dµ 0 1-2 n ≤ B + λ 0 + 4(n -1) n -2 1 r 2 ˆB(x,R+r) u 2 dµ 0 + ˆB(R+r) Scal g f n 2 + dµ f 2 n ˆB(x,R+r) (ξu) 2n n-2 dµ 0 1-2 n . (11)
We obtain hence that for a constant C that depends only on g 0 , R 0 and n:

(12) ˆB(x,R) u 2n n-2 dµ 0 1-2 n ≤ C δr 2 ˆB(x,R+r) u 2 dµ 0 .
Therefore, if ε ≥ 2, using again Hölder inequality and setting r = R ≤ R 0 /2 we get

ˆB u 2n n-2 dµ 0 n-2 n ≤ C δR 2 ˆ2B u ε dµ 0 2 ε µ 0 (B) ε-2 ε
which gives Lemma 3.9. Assume now that ε < 2 and let us go to the second stage of the proof. We will use a trick of P. Li and R. Schoen [START_REF] Li | L p and mean value properties of subharmonic functions on Riemannian manifolds[END_REF]Theorem 2.1] in order to obtain the wanted conclusion. Assuming now that r ≤ R ≤ R 0 /2, and using Hölder inequality with exponent

α = 1 + 4 (2 -ε)(n -2) we obtain ˆB(x,R) u 2n n-2 dµ 0 1-2 n ≤ C δr 2 ˆB(x,R+r) u ε dµ 0 1-1 α ˆB(x,R+r) u 2n n-2 dµ 0 1 α . Let us define now λ = 1 α n n-2 , M (R) = ˆB(x,R) u 2n n-2 dµ 0 1-2 n and I = ˆB(x,2R) u ε dµ 0 1-1 α .
Notice that λ ∈ (0, 1) since

1 λ = 1 + 2ε n(2 -ε)
. Equation [START_REF] Carron | The Huber theorem for non-compact conformally flat manifolds[END_REF] shows hence that

M (R) ≤ CI δr 2 M (R + r) λ . Iterating this equation with r 1 + r 2 + • • • + r k ≤ R/2 leads to M (R) ≤ CI δ 1+λ+•••+λ k-1 k ℓ=1 1 r 2λ ℓ-1 ℓ M (R + r 1 + r 2 + • • • + r k ) λ k .
If we choose r ℓ = R2 -ℓ-2 , with β = 1 1-λ , a little bit of arithmetic gives that

M (R) ≤ C 1 R 2β I β . It follows that B(x,R) u 2n n-2 dµ 0 ≤ C B(x,2R) u ε dµ 0 2n ε(n-2)
which concludes the proof of Lemma 3.9.

Let us now conclude the proof of Theorem 3.7. Let ε > 0 be given by Lemma 3.8, and set p = 2n ε(n-2) . Let us write w = u 2n n-2 = e nf . It follows from Lemmas 3.8 and 3.9 that there exists C = C(n, g 0 , R 0 , δ, Λ, ε) > 0 such that for all g 0 -balls B with radius less than R 0 /2,

1 2 B wdµ 0 B w -1 p dµ 0 p ≤ C.
Hence for any g 0 -ball B with radius less than R 0 /4:

B wdµ 0 B w -1 p dµ 0 p ≤ θ p B wdµ 0 2B w -1 p dµ 0 p ≤ θ p C,
where θ is the doubling constant for the volume measure g 0 . This is precisely characterization ii) of A ∞ weights, cf Definition 3.1.

STRONG A ∞ WEIGHTS AND BI-H ÖLDER COMPACTNESS

We now introduce so-called strong A ∞ weights, also known as metric doubling measures, which are special cases of A ∞ weights with strong geometric controls, such as Sobolev, Poincaré and isoperimetric inequalities.

We could not produce examples of sequences of conformal deformations satisfying the hypotheses of Theorem 1.2 which would not be uniformly strong A ∞ . Our Theorem 1.3 shows that any conformal deformation is actually uniformly strong A ∞ provided the L n/2 norm of the scalar curvature on small balls is pinched enough, which explains this fact.

4.1. Definitions and basic facts. Let (M, g 0 ) be a closed Riemannian manifold, µ 0 its Riemannian volume. If g f = e 2f g 0 is a metric conformal to g 0 , we continue to denote by µ f and by d f the corresponding Riemannian volume and distance respectively. Definition 4.1. Let g f = e 2f g 0 be a conformal deformation of g 0 . We say that the volume density e nf is a strong-A ∞ weight with respect to the Riemannian metric g 0 if there are positive constants η, θ such that

(1) x ∈ M, r ≤ η ⇒ µ f (B(x, 2r)) ≤ θµ f (B(x, r)). (2) x, y ∈ M, d 0 (x, y) ≤ η ⇒ d f (x, y) n θ n ≤ µ f (B(x, d 0 (x, y))) ≤ θ n d f (x, y) n .
In above definition, B(x, r) is a geodesic ball of radius r for the fixed d 0 = d g0 distance. Notice that the strong A ∞ condition is invariant under scaling: if dµ f = e nf dµ 0 is a strong-A ∞ weight with constants (θ, η) then for every h > 0, h n dµ f is also a strong-A ∞ weight with same constants (θ, η).

These strong A ∞ weights were introduced in [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF], in relation with the quasiconformal jacobian problem. The jacobian of a quasiconformal map is a strong A ∞ weight, as is any A 1 -weight (see Definition 3.1 ii)). A strong A ∞ weight with constants (θ, η) is an A ∞weight with constants only depending on θ, η, see Proposition 3.4 of [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF]. Nevertheless, not all A ∞ weights are strong A ∞ . Various characterizations of strong A ∞ weights can be found in [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF] and [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF], together with examples and details about their relationships with the quasiconformal jacobian problem and the bi-Lipschitz embedding problem.

It is easy to see that if the conditions in definition 4.1 are satisfied for some θ, η then for any other η ′ ≥ η there is a constant θ ′ depending on (θ, η) and on the geometry of g 0 such that the same conditions are satisfied with constant (θ ′ , η ′ ). We also remark that the doubling condition on µ f implies that if m ∈ M is a point such that

ρ = d 0 (m, x) = d 0 (m, y) = 1 2 d 0 (x, y) ≤ η 2
then the balls B(m, ρ), B(x, d 0 (x, y)), B(y, d 0 (x, y)) have comparable µ f mesure. Note that when d 0 (x, y) ≤ 1 2 inj(M, g 0 ), there is a unique point m ∈ M such that

ρ = d 0 (m, x) = d 0 (m, y) = 1 2 d 0 (x, y).
The ball B(m, ρ) will be denoted B x,y . We obtain thus the following alternative and equivalent definition of strong A ∞ weight.

Definition 4.2. If θ, η are positive number such that η ≤ 1 4 inj(M, g 0 ), we say that the volume density e nf associated to the metric

g f = e 2f g 0 is a strong-A ∞ weight (with respect to g 0 ) with constant (θ, η) if (1) x ∈ M, r ≤ η ⇒ µ f (B(x, 2r)) ≤ θµ f (B(x, r)). (2) x, y ∈ M, d 0 (x, y) ≤ 2η ⇒ d f (x, y) n θ n ≤ µ f (B x,y ) ≤ θ n d f (x, y) n .
It follows from Lemma 3.5 that the left inequality in property (2) in definition 4.1 is satisfied for all A ∞ weights. The specificity of strong A ∞ weights relies hence in the other inequality.

Let g f = e 2f g 0 be a conformal deformation of g 0 such that its volume density e nf is a strong-A ∞ weight (with respect to g 0 ) with constants (θ, η). We are going to show several basic controls on the geometry of (M, g f ).

Geometry of g f balls.

In this subsection, we show that the strong A ∞ condition implies that the geodesic ball for g f and g 0 are comparable. Let B f (x, r) denote the g f -geodesic ball of center x and radius r.

In this section and in what follows we use the following notation: R(x, r) := µ f (B(x, r))

1 n ,
where, as before, B(x, r) is the g 0 geodesic ball of center x and radius r.

Lemma 4.3.

There exists κ = κ(η, θ) > 0 such that for any r ≤ η/κ :

B f (x, R(x, r/κ)) ⊂ B(x, r) ⊂ B f (x, R(x, κr)).
Proof. The doubling condition implies a reverse doubling estimate (see [START_REF] Grigor'yan | Stability results for Harnack inequalities[END_REF]Lemma 2.10]): there are positive constants α, ω depending only on η and θ such that [START_REF] Chang | Lack of compactness in conformal metrics with L d/2 curvature[END_REF] x

∈ M, 0 < s < r ≤ η ⇒ ω -n s r n α ≤ µ f (B(x, s))) µ f (B(x, r))) ≤ ω n s r nα .
The doubling condition also implies that the µ f -measure of g 0 -geodesic spheres is zero ( [START_REF] Colding | Liouville theorems for harmonic sections and applications[END_REF][START_REF] Tessera | Volumes of spheres in doubling measures metric spaces and groups of polynomial growth[END_REF]), hence the function r → R(x, r) is continuous and strictly increasing, in particular we have R(x, s) < R(x, r) ⇐⇒ s < r.

We have for d 0 (x, y) ≤ η

(14) θ -1 ≤ d f (x, y) R(x, d 0 (x, y)) ≤ θ. and x ∈ M, 0 < s < r ≤ η ⇒ ω -1 s r 1 α ≤ R(x, s) R(x, r) ≤ ω s r α .
With κ := (ωθ) 1 α , and assuming without loss of generality that κ > 1, we get for any r ≤ η/κ : θR(x, r) ≤ R(x, κr) and R(x, r/κ) ≤ θ -1 R(x, r).

Since by ( 14), we have

B f (x, R(x, r)/θ) ⊂ B(x, r) ⊂ B f (x, θR(x, r)) we get B f (x, R(x, r/κ)) ⊂ B(x, r) ⊂ B f (x, R(x, κr)).
Note that the two radius R(x, r/κ) and R(x, κr) are uniformly comparable:

ω -1 κ 2 α ≤ R(x, κr) R(x, r/κ) ≤ ωκ 2α .
4.1.3. Volume doubling for (M, d f , µ f ). As we mentioned above, if a weight is strong-A ∞ , it is A ∞ . We proved that A ∞ weights are doubling for the balls of the background metric. We show now that the metric measure space (M, d f , µ f ) is doubling (for g fgeodesic balls) with constants only depending on η and θ.

Lemma 4.4. There exists δ = δ(η, θ) > 0 and k = k(η, θ) > 0 such that for all R ∈ (0, δ),

µ f (B f (x, 2R)) ≤ 2 k µ f (B f (x, R)), where B f (x, R) is the d f -geodesic ball of radius R and center x.
Proof. Let D be the diameter of (M, g 0 ), then according to [11, Lemma 3.10], the volume doubling condition for (M, d 0 , µ f ) implies that for any x ∈ M and any r ≤ η :

(15) µ f (M ) ≤ θ 50+50 D r µ f (B(x, r)). Hence R(x, s) ≤ θ -50-50 D r µ f (M ) 1 n ⇒ s ≤ r.
Now we have for r ≤ η/κ :

µ f (B f (x, R(x, r))) ≤ µ f (B(x, κr)) ≤ θ k µ f (B(x, 2 -k rκ)) ≤ θ k µ f (B f (x, R(x, 2 -k rκ 2 )) But R(x, 2 -k rκ 2 ) ≤ ω 2 -k κ 2 α R(x, r) Chosing k such that ω 2 -k κ 2 α ≤ 1 2 we get µ f (B f (x, R(x, r))) ≤ θ k µ f (B f (x, R(x, r)/2)).

Hence with

δ := θ -50-50κ D η µ f (M ) 1 n ,
this eventually implies that

R ≤ 2δ ⇒ µ f (B f (x, 2R)) ≤ 2 k µ f (B f (x, R)).
4.1.4. Estimate of the diameter of (M, g f ). We are going to show that the diameter of (M,

g f ) is comparable to µ f (M ) 1 n .
Lemma 4.5. There is a constant C depending only on g 0 , θ, η such that

C -1 (µ f (M )) 1 n ≤ diam(M, g f ) ≤ C (µ f (M )) 1 n .
Proof. The upper bound for diam(M, g f ) is an immediate consequence of Lemma 3.5.

Let us show the lower bound.

It is always possible to assume moreover that η ≤ D. Then there is always two points x, y ∈ M with d 0 (x, y) = η and we have

diam(M, g f ) ≥ d f (x, y) ≥ θ -1 (µ f (B(x, η))) 1 n ,
therefore by [START_REF] Costea | Strong A∞-weights and scaling invariant Besov capacities[END_REF], we get

diam(M, g f ) ≥ C(θ, η, D) (µ f (M )) 1 n .

Bi-Hölder compactness for sequence of Strong

A ∞ conformal metric. Let (g k = e 2f k g 0 ) k∈N be a sequence of conformal deformations of g 0 such that the weights e nf k are strong-A ∞ weight with uniform constant (θ, η). As mentioned earlier, it follows from [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF] that all the g k are uniformly A ∞ . If the volumes of (M, g k ) are uniformly bounded, Proposition 3.6 implies then that the sequence of metric spaces (M, d f k ) k∈N has a subsequence which converges in Gromov-Hausdorff topology to some limiting space (M ∞ , d ∞ ). Strong A ∞ weights will give the following stronger convergence result. Proposition 4.6. Let (g k = e 2f k g 0 ) k∈N be a sequence of Riemannian metrics conformal to g 0 such that the weights e nf k are strong-A ∞ weights with uniform constant (θ, η).

i) If there are positive constants v, V such that ∀k : v ≤ µ f k (M ) ≤ V , then there is a subsequence that converges in the C α and Gromov-Hausdorff topology to

(M, d ∞ ), where d ∞ is a distance on M which is bi-Hölder equivalent to d 0 . ii) If lim k→∞ µ f k (M ) = 0, then the sequence of metric spaces ((M, d f k )) k converges in the Gromov-Hausdorff topology to a point. iii) If there is a positive constant v such that ∀k : v ≤ µ f k (M ), then for any p ∈ M ,
there is a subsequence of the sequence of pointed metric spaces ((M, d f k , p)) k that converges in the pointed Gromov-Hausdorff topology to a pointed metric space.

Proof. We will only show i); the other cases follow from a straightforward scaling argument. As soon as e nf is a strong A ∞ with constants η, θ, we have seen that if

d 0 (x, y) ≤ η then θ -1 R(x, d 0 (x, y)) ≤ d f (x, y) ≤ θR(x, d 0 (x, y)).
Moreover by ( 13) we have

ω -1 d 0 (x, y) η 1 α µ f (B(x, η)) 1 n ≤ R(x, d 0 (x, y)) ≤ ω d 0 (x, y) η α µ f (B(x, η)) 1 n .
Hence if v ≤ µ f (M ) ≤ V , there is a constant λ depending only on v, V, θ, ω, η, α such that 1 λ (d 0 (x, y))

1 α ≤ d f (x, y) ≤ λ (d 0 (x, y)) α .
Hence the identity map

Id : (M, d f ) → (M, d 0 )
is uniformly bi-Hölder continuous Therefore a uniform strong A ∞ estimate for the (e nf k ) k∈N together with a uniform control on the volume 0 < v ≤ µ f k (M ) ≤ V implies that the sequence (d f k ) is precompact in the Hölder topology and for any converging subsequence, the limiting map d ∞ is a distance on M which is uniformly bi-Hölder equivalent to d 0 .

4.1.6. Sobolev, Poincaré and isoperimetric inequalities. Let us now mention some other important geometric facts about strong A ∞ weights shown in [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF].

Theorem 4.7 [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF]). Let (M, g 0 ) be a compact smooth Riemannian manifold, and let ω = e nf be a strong A ∞ weight with respect to g 0 with constants η and θ, and let g f = e 2f g 0 = ω 2 n g 0 . Then i) There is a constant θ = θ(g 0 , η, θ) such that for any x ∈ M and any g f -geodesic B(x, r) of radius less than diam g f (M ):

θ -1 r n ≤ µ f (B(x, r)) ≤ θr n .
ii) There is a constant γ = γ(g 0 , η, θ) > 0 such that for any smooth domain Ω ⊂ M with µ f (Ω) ≤ 1 2 µ f (M ) , we have

γµ f (Ω) n-1 n ≤ µ f (∂Ω).
iii) There is a constant λ = λ(g 0 , η, θ) such that for any g f -geodesic B(x, r) of radius less than diam g f (M ): and any ϕ ∈ C 1 (B(x, r)) with ´B(x,r) ϕdµ f = 0 then

ˆB(x,r) |ϕ|dµ f ≤ λr ˆB(x,r) |dϕ| g f dµ f
Sobolev inequalities for (M, g f ) are consequences of doubling and Poincaré inequality, see [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF]. 4.2. Q-curvature and Strong A ∞ bounds. Let us finish this section by presenting the results of S. Brendle [START_REF] Brendle | Global existence and convergence for a higher order flow in conformal geometry[END_REF] and Yi Wang [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF] which have inspired our analysis of conformal metrics with L n/2 pinching of the scalar curvature via A ∞ weights. These works are devoted to the study of conformal deformations with L 1 bounds for the Branson Q-curvature, which we present now.

Let (M, g) be a Riemannian manifold of dimension n = 4. The Branson Q-curvature of (M, g) is defined by

Q g = 1 12 ∆ g Scal g + 1 4 Scal 2 g -3 |E g | g ,
where E g is the traceless part of the Ricci tensor. This Q-curvature has a natural conformal invariance: if g f = e 2f g 0 , then

(16) 2e 4f Q g f = P g0 f + 2Q g0 ,
where P g0 is the Paneitz operator on (M, g 0 ), which we shall not define here (see for instance [START_REF] Brendle | Global existence and convergence for a higher order flow in conformal geometry[END_REF]). This equation is the analogous for the Q-curvature or the Yamabe equation ( 6) for the scalar curvature. The existence of a Paneitz operator and a Q-curvature satisfying the conformal covariance property [START_REF] Chang | Lack of compactness in conformal metrics with Ld/2 curvature[END_REF] has been generalized to all even dimension by Fefferman and Graham in [START_REF] Fefferman | Q-curvature and Poincar metrics[END_REF]. Note that the leading term of P g0 is always (∆ g0 ) n 2 , and P g0 = (∆ g0 ) n 2 when g 0 is a flat metric.

The following result was shown in [START_REF] Brendle | Global existence and convergence for a higher order flow in conformal geometry[END_REF], Proposition 1.4. Theorem 4.8 [START_REF] Brendle | Global existence and convergence for a higher order flow in conformal geometry[END_REF]). Let (S n , g S ) be the standard n-sphere and let C, δ > 0 and R 0 ∈ (0, diam(S n )) be fixed. Let (g k = e 2f k g S ) k∈N be a sequence of conformal deformations of g s with constant volume such that:

• ˆSn Q 2 g k dvol g k ≤ C; • for all x ∈ S n , ˆB(x,R0) |Q g k | dvol g k ≤ 1 2 (n -1)!σ n -δ,
where B(x, R 0 ) is the g S -geodesic ball of radius x and center x. the critical constant in Brendle theorem is half of the total Q-curvature of the sphere. It was later shown by Wang in [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF] that under an analogous L 1 pinching condition for the Q-curvature, conformal deformations of the euclidean metric on R n are uniformly strong A ∞ weights.

Then (f k ) k∈N is bounded in H n = W
Theorem 4.9 (Wang ( 2013)). Let C, δ > 0 be fixed and let (g f = e 2f eucl) k∈N be a conformal deformation of the euclidean plane (R n , eucl), which is normal metrics, and such that:

• ˆM Q g f dµ f ≤ C; • ˆM Q + g f dµ f ≤ 1 2 (n -1)!σ n -δ.
Then the Riemannian volume density (e nf ) k∈N is a strong A ∞ weight with respect to the standard Lebesgue measure on R n , with constants only depending on n, C and δ.

A normal metric on R n is a Riemannian metric whose behaviour at infinity is not too pathological, we refer to [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF] for a precise definition. For instance, any smooth metric whose scalar curvature is non-negative at infinity is normal.

The previous theorem is a reformulation of Theorems 3.2 and 4.1 of [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF]. The parallel between Brendle and Wang's assumptions and the assumptions of our Theorem 1.3 is obvious; indeed our approach was inspired by Wang's proof. Note that the constant 1 2 (n-1)!σ n is optimal to get a strong A ∞ weight, as can easily be shown by considering a sequence of metrics built from removing a ball in R n and gluing on its boundary a very long cylinder with a round cap. Such sequence (even smoothed) cannot be uniformly strong A ∞ , since it has no uniform isoperimetric inequality.

The proof of Wang can be localized using the same approach as we do in this paper, which gives following result. Theorem 4.10. Let (M, g 0 ) be a compact smooth manifold, and C, δ > 0 and R 0 ∈ (0, diam(M, g 0 )) be fixed. Let (g f = e 2f g 0 ) k∈N be a sequence of conformal deformations of g 0 , such that:

• ˆM Q g f dµ f ≤ C; • for all x ∈ M , ˆB(x,R0) Q + g f dµ f ≤ 1 2 (n -1)!σ n -δ.
Then the Riemannian volume density (e nf ) k∈N is a strong A ∞ weight with respect to g 0 , with constants only depending on (M, g 0 ), C and δ.

We will not detail here the proof of this result, which is a combination of our arguments with those of [START_REF] Wang | The isoperimetric inequality and Q-curvature[END_REF].

L n/2 PINCHING FOR THE SCALAR CURVATURE AND STRONG A ∞ WEIGHTS

In Section 4 we explained that a uniform strong A ∞ control coupled with uniform volume bounds has important geometric consequences, which were gathered in Corollary 1.4. This section is devoted to the proof of Theorem 1.3, which is our second main result. In order to motivate the sequel of our study, we start with the proof of Theorem 1.3 assuming Theorems 5.1 and 5.7 which will be stated and proved in Sections 5.2 and 5.3 respectively. 5.1. Proof of Theorem 1.3. Let g f = e 2f g 0 be a conformal deformation such that for some Λ 0 small enough (only depending on g 0 ) and for some R 0 > 0, we have

∀x ∈ M, ˆB(x,R0) Scal g f n 2 dµ f ≤ Λ 0 .
By Yamabe's equation [START_REF] Bonk | The quasiconformal Jacobian problem[END_REF], writing e 2f = u 4 n-2 , we have

4(n -1) n -2 ∆ g0 u = Scal g f u 4 n-2 u -Scal g0 u.
This is a Schrödinger type equation of the form ∆ g0 u + q f u = 0, with

q f = n -2 4(n -1) Scal g0 -Scal g f e 2f .
Moreover, for all x ∈ M ,

q f L n 2 (B(x,R0),g0) ≤ C(g 0 )R 2 0 + ´B(x,R0) Scal g f n 2 dvol g f 2 n ≤ C(g 0 )R 2 0 + Λ 0 .
Hence if one chose Λ 0 to be smaller that the half of the ǫ of the Theorem 5.1 , then for R 0 small (so that C(g 0 )R 2 0 ≤ ǫ/2) the Schrödinger operator ∆ g0 + q f satisfies the hypothesis of the Theorem 5.1 and since u is a positive smooth solution of (∆ g0 + q f )u = 0, the lowest eigenvalue of ∆ g0 + q f is necessary zero, with eigenfunction u. By Theorem 5.1, we can decompose

f = h + w, where h ∈ W 2, n 2 (M, g 0 ) and w ∈ C α (M, d 0 ), with sup x,y∈M |w(x) -w(y)| d 0 (x, y) α ≤ C(g 0 , R 0 , Λ 0 )
and ( 17) [START_REF] David | Strong A∞ weights, Sobolev inequalities and quasiconformal mappings[END_REF] with Theorem 5.7, there exists η = η(g 0 , R 0 , Λ 0 ) and θ = θ(g 0 , R 0 , Λ 0 ) such that e nh is a strong A ∞ weight with respect to (M, g 0 ) with constants η and θ. Since ww(x 0 ) is uniformly bounded from above and below, and since being Strong A ∞ weight is a scale invariant condition, this implies that e nf = e n(h+w-w(x0)) e nw(x0) is also a strong A ∞ weight with constants η and θ. Moreover, by Proposition 5.9, for all α ∈ (0, 1), the distances d f and d 0 are α-bi-Hölder with constants only depending on g 0 , R 0 , Λ 0 and α, which concludes the proof of Theorem 1.3.

dh L n + ∆h L n 2 ≤ C(g 0 , R 0 , Λ 0 ). Let x 0 be a chosen base point in M . Since w is bounded in the C α topology induced by d 0 , there exists C > 0 such that for all x ∈ M , |w(x) -w(x 0 )| ≤ Cdiam(M, g 0 ) α . Now, combining

5.2.

On the ground state of Schrödinger operators with critical potential. The purpose of this section is to give a regularity result for the first eigenfunction of a Schrödinger operator whose potential has small L n/2 norm on small balls, which is the first key step in the proof of Theorem 1.3. The results presented in this paragraph are valid for any compact Riemannian manifold (M, g) and of independent interest. Theorem 5.1. Let (M n , g) be a closed Riemannian manifold of dimension n > 2. There are constants ǫ, α > 0 depending only of (M n , g) such that if V ∈ L n 2 satisfies: [START_REF] Akutagawa | Yamabe metrics of positive scalar curvature and conformally flat manifolds[END_REF] the lowest eigenvalue of the Schrödinger operator ∆ -V is zero;

(2) there is some ρ > 0 such that for any x ∈ M :

ˆB(x,ρ) |V | n 2 < ǫ.
Then for all positive

C 2 solution ϕ > 0 of ∆ϕ -V ϕ = 0, there are f ∈ W 2, n 2 and w ∈ C α (M ) such that ϕ = e f +w .
Moreover, writing

I := sup x ˆB(x,ρ) |V | n 2 2 n
we have

w Ċα := sup x,y∈M |w(x) -w(y)| d 0 (x, y) α ≤ C(g, ρ, I) and df L n + ∆f L n 2 ≤ C(g, ρ)I.
This theorem is a key step in showing that conformal deformations whose scalar curvature has small L n/2 norm on small balls are strong A ∞ weights.

In order to prove Theorem 5.1, we need some preliminary tools. Let us start by recalling the following classical consequence of the fixed point theorem.

Proposition 5.2. Let (X, • ) be a Banach space, κ ∈ (0, 1). If S : B(0, ρ) → X is a κ-Lipschitz map with S(0) + κρ < ρ then there is a unique x ∈ B(0, ρ) such that S(x) = x. Moreover we have x ≤ S(0) 1κ .

Let A(g) be the best constant in the following functional inequality on the closed Riemannian manifold (M, g):

(18) ∀ϕ ∈ C ∞ (M ) : dϕ L n ≤ A(g) ∆ϕ L n 2 . Proposition 5.3. Let (M, g) be a compact smooth Riemannian manifold such that Ric g ≥ -(n -1)k 2 g, then the best constant A(g) in (18) satisfies A(g) ≤ e c(n)(1+k diam(M,g)) diam(M, g) vol 1 n (M, g) .
The above estimate on A(g) can be shown using a lower bound on the Ricci curvature, and upper bounds on the diameter and volume of (M, g); it follows from results of S. Gallot [START_REF] Gallot | Inégalités isopérimétriques et analytiques sur les variétés riemanniennes[END_REF] and of D. Bakry [3, Théorème 4.1, Lemme 4.2 and Corollaire 4.3]. We will not show it here, since we will not need this expression.

Let us now show the following property of the ground state of the Schrödinger operator

∆ -V . Proposition 5.4. Let V ∈ L ∞ (M ) ∩ L n 2 (M )
and let λ 0 be the bottom of the spectrum of the Schrödinger operator ∆ -V . If

8A(g) 2 V L n 2 < 1 then for any ϕ > 0 satisfying ∆ϕ -V ϕ = λ 0 ϕ, we have d log ϕ L n ≤ 2A(g) V L n 2 .
Proof. We remark that if ϕ = e u satisfies ∆ϕ -V ϕ = λ 0 ϕ, then we have ∆u -|du| 2 = V + λ 0 . Moreover up to a scaling constant, there is a unique non trivial solution to the equation ∆ϕ -V ϕ = λ 0 ϕ. We introduce the Banach space

W 1,n ⊥ := {v ∈ W 1,n (M ), ˆM v = 0} endowed with the norm v ⊥ = dv L n .
We introduce the operator S :

W 1,n ⊥ → W 1,n ⊥ defined by S(v) = ∆ -1 V + ∆ -1 |dv| 2 .
where for all f ∈ L n 2 , we have denote u := ∆ -1 f the solution of the equation

∆u = f -f M ´M u = 0.
By definition of A(g), we have

S(0) ⊥ ≤ A(g) V L n 2 , and S(u) -S(w) ⊥ ≤ A(g) u -w ⊥ ( u ⊥ + w ⊥ ) .
As a consequence the restriction of S to the ball

B(0, ρ) ⊂ W 1,n ⊥ , is 2A(g)ρ-Lipschitz and if (19) A(g) V L n 2 + 2A(g)ρ 2 < ρ then there is a unique v ∈ B(0, ρ) with S(v) = v and v ⊥ ≤ A(g) V L n 2 1 -2A(g)ρ .
Choosing ρ = (4A(g)) -1 , the condition ( 19) is satisfied as soon as

A(g) V L n 2 < (8A(g)) -1 . We have obtained hence a function v ∈ W 1,n ⊥ such that S(v) = v and v ⊥ ≤ 2A(g) V L n 2 . Therefore there is a constant c such that ∆v -|dv| 2 = V + c, therefore ∆e v -V e v = ce v .
Since the only positive eigenfunction of ∆ -V is associated to λ 0 , we have c = λ 0 , e v = kϕ for some k > 0 and ||d log ϕ||

L n = ||v|| ⊥ ≤ 2A(g) ||V || L n/2 .
Let us now define the Sobolev constant β = β(g 0 ) > 0 of (M, g 0 ) to be the best constant in the inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] ∀ϕ ∈ C ∞ (M ) :

β ϕ 2 L 2n n-2 ≤ dϕ 2 L 2 + ϕ 2 L 2 .
Note that this Sobolev inequality implies that

β ≤ Vol(M, g 0 ) 2 n .
We also have that for any δ ∈ (0, 1):

∀ϕ ∈ C ∞ (M ) : δ 2 β ϕ 2 L 2n n-2 ≤ ˆM |dϕ| 2 g0 + δ 2 |ϕ| 2 dµ 0 .
Hence we can easily prove the following lemma:

Lemma 5.5. If W ∈ L n 2 satisfies W L n 2 ≤ βδ 2 , then the Schrödinger operator ∆ + δ 2 + W is non negative.
Proof. The Hölder inequality implies that if ϕ ∈ C 1 (M ):

-ˆM W ϕ 2 ≤ W L n 2 ϕ 2 L 2n n-2 ≤ δ 2 β ϕ 2 L 2n n-2 ≤ ˆM |dϕ| 2 g0 + δ 2 |ϕ| 2 dµ 0
Lemma 5.6. Assume that x 0 ∈ M and r 0 are chosen so that

Vol B(x 0 , r 0 ) ≤ β 2 n 2
.

If q ∈ L n 2 is such that supp(q) ⊂ B(x 0 , r 0 ) and q L n 2 ≤ β then there is a constant c 0 satisfying :

|c 0 | ≤ 2 β q L n 2
such that the lowest eigenvalue of the Schrödinger operator

∆ + q + c 0 1 M\B(x0,r0)
is zero.

Proof. Let λ 0 (c) be the lowest eigenvalue of the Schrödinger operator

∆ + q + c 1 M\B(x0,r0) .
Using the constant function, we get that λ 0 (c) ≤ 0 as soon as

ˆM q + c Vol g0 (M \ B(x 0 , r 0 )) ≤ 0.
Moreover, Vol B(x 0 , r 0 ) ≤ Vol(M,g0)

2

, hence for c -= -2 Vol(M,g0) ´M |q|, we have λ 0 (c -) ≤ 0.

Using Lemma 5.5 for W = qc 1 B(x0,r0) , we also see that if c > 0 satisfies qc L Using the work of O. A. Ladyzhenskaya and N. N. Ural'tseva [START_REF] Ladyzhenskaya | Estimates for the Hölder constant for functions satisfying a uniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients[END_REF] and of M.V. Safonov [START_REF] Safonov | Non-divergence Elliptic Equations of Second Order with Unbounded Drift[END_REF], we know that ψ i is Hölder continuous. There is some α = α(g 0 , ρ, ǫ) such that ψ i Ċα (B(xi,3ρ/4)) := sup

x,y∈B(xi,3ρ/4))

|ψ i (x) -ψ i (y)| d 0 (x, y) α ≤ C(g 0 , ρ, I) inf x∈B(xi,3ρ/4)) ψ i (x).
We let δ i = inf B(xi,3ρ/4) ψ i and ψ i = δ i e wi and ϕ i = e fi δ i and f i = ffl B(xi,3ρ/4) f i . We have w i L ∞ (B(xi,3ρ/4)) + w i Ċα (B(xi,3ρ/4)) ≤ C(g 0 , ρ, I)

and df i L n + ∆f i L n 2 ≤ C(g 0 )I.

Moreover, by Sobolev inequality

f i -f i L n ≤ C(g 0 )I.
Note that on all ball B(x i , ρ), we have [START_REF] Grigor'yan | The heat equation on noncompact Riemannian manifolds (Russian)[END_REF] ϕ = ϕ i ψ i = e wi e fi .

Let us fix now a partition of unity (χ i ) i∈{1,...,N } with suppχ i ⊂ B(x i , 3ρ/4) and χ i = 1 on B(x i , ρ/4). This partition of unity only depends on (M, g 0 ). Let us define

f := i χ i f i -f i and w = i χ i (w i + f i ).
The functions f and w are well defined on M and it follows from ( 21) that ϕ = e f e w . We have moreover d log f L n + ∆f L

ˆM |d log ϕ| 2 dµ 0 ≤ ˆM |V |dµ 0 ≤ (µ 0 (M )) 1-2 n V L n 2 ≤ C(g 0 )I.
Using the Poincaré inequality, one conclude that for c = ffl M log(ϕ)dµ 0 :

f + w -c L 2 ≤ C(g 0 )I.
On each ball B(x i , ρ/4), the oscillation of w i = wf i are controlled hence one get that

f i -c ≤ C(g 0 ) f i -c L 2 (B(xi,ρ/4)) ≤ C(g 0 , I).
Hence w = χ i w i + f i is bounded in Ċα (M ).

5.3.

Strong A ∞ weights from W 1,n bounds. In this section, we explain how the proof of Theorem 3.1 of [START_REF] Bonk | The quasiconformal Jacobian problem[END_REF] can be adapted for closed Riemannian manifolds, which gives the following result. This is the second key step in the proof of Theorem 1.3. Theorem 5.7. Let (M, g 0 ) be a closed Riemannian manifold. If f : M → R is such that df L n ≤ I, for some I. Then the weight e nf is a strong A ∞ weight with respect to g 0 , with constants (θ, η) where η depends only on g 0 and θ depends only on g 0 and I.

Note that, unlike Theorem 3.1 of [START_REF] Bonk | The quasiconformal Jacobian problem[END_REF], our statement and our proof are scale invariant.

5.3.1.

Presentation of the setting. The estimates will depend on several geometric and analytic estimates. We continue to denote by µ 0 the Riemannian volume measure and by d 0 the Riemannian distance associated to g 0 . There is some η > 0 and positive constants θ, γ, C (that can be chosen depending only on the dimension n) such that for any ball B ⊂ M with radius r(B) ≤ 2η we have i) as already mentioned in (8), θ -1 r(B) n ≤ µ 0 (B) ≤ θr(B) n ; ii) as shown in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], Lemma 7.16, for any Lipschitz function ϕ : B → R and any x ∈ B:

|ϕ(x) -ϕ B | ≤ C ˆB |dϕ| g0 (z) d n-1 0 (x, z) dµ 0 (z);
iii) the following Adams-Moser-Trudinger inequality: for any Lipschitz function ϕ : B → R and any x ∈ B:

B e γ|ϕ(x)-ϕ B | dϕ L n (B) n n-1 dµ 0 (x) ≤ C.
For the Euclidean space R n , this inequality is due to N. Trudinger [START_REF] Trudinger | On imbeddings into Orlicz spaces and some applications[END_REF]. According to [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Lemma 7.13], the first two geometric control i-ii) implies the Trudinger inequality. We will use several times the following integration by parts formula

ˆB(x,R)\B(x,r) ϕ(d 0 (x, y))f (y)dµ 0 (y) = ϕ(R) ˆB(x,R) f (y)dµ 0 (y) -ϕ(r) ˆB(x,r) f (y)dµ 0 (y) - ˆR r ϕ ′ (r)
ˆB(x,r) f (y)dµ 0 (y)dr.

5.3.2.

Estimate on µ f (B). We are going to prove that µ f is an A p weights for every p > 1 using characterization ii) in Definition 3.1. Let B(o, ρ) ⊂ M be such that its radius satisfies r(B) ≤ η. We have for any x ∈ B and any λ ∈ R:

λ|f (x) -f B | ≤ n -1 n γ |f (x) -f B | df L n (B) n n-1 + 1 n |λ| γ df L n (B) n .
Hence, using Adams-Moser-Trudinger inequality, we get This easily implies that for any p > 1 there is a constant C depending on g 0 , I, p such that ( 23)

B e nf dµ 0 B e -n p-1 f dµ 0 p-1 ≤ C.
Hence, e nf is an A p weight, in particular dµ f is a doubling measure. Moreover, by Lemma 3.5, this implies that there exists a constant C = C(g 0 , I) > 0 such that

d 0 (x, y) ≤ 10η ⇒ d f (x, y) ≤ C (µ f (B x,y )) 1 n .
5.3.3. Distance estimate. To show that dµ f is a strong A ∞ weight, we are only left with showing that for all x, y ∈ M and for some C > 0,

d g f (x, y) ≥ Cµ f (B x,y ) 1 n .
In this purpose, we will need the following lemma:

Lemma 5.8. For any ǫ > 0 there is a constant C depending on g 0 , I and ǫ > 0 such that for any ball B ⊂ M with r(B) ≤ η there is a set

E ⊂ B such E ⊂ α B α with α r(B α ) ≤ ǫr(B) and x ∈ B \ E ⇒ |f (x) -f B | ≤ C.
Proof. Assume that x ∈ B is such that for any r ∈ (0, 2r(B)) we have With Hölder inequality, one has

ˆB(x,r) |df | n ≤ λ r r(B) then using V (x,
V (x, r) ≤ θ 1-1 n r n-1 ˆB(x,r) |df | n 1/n ≤ θ 1-1 n r n-1 λ r r(B) 1/n .
Hence there is a constant depending only of g 0 such that

|f (x) -f B | ≤ Cλ 1 n .
If now E is the subset of point x where there is a r x ∈ (0, 2r(B)) such that

ˆB(x,rx) |df | n > λ r x r(B)
Then using Vitali covering lemma, one gets a covering E ⊂ ∪B α where

B α = 5B(x α , r α ) such that ˆB(xα,rα) |df | n > λ r α r(B)
and the balls B(x α , r α ) are disjoints. In particular this implies that

r(B α ) = 5r α ≤ 5r(B) λ ˆB(xα,rα) |df | n ≤ 5r(B) λ ˆM |df | n .
Choosing λ = 5 ǫI n , we get the desired result. we can find

E ⊂ 3B such that E ⊂ α B α with α r(B α ) ≤ ǫ 3 2 ρ and x ∈ 3B \ E ⇒ |f (x) -f B | ≤ C.
We have

L g f (γ) ≥ L g f ( γ| [0,τ ] ) ≥ L g f ( γ| [0,τ ] \ E) ≥ e -C e f B L g0 γ| [0,τ ] \ E ≥ e -C e f B κ -1 L eucl γ| [0,τ ] \ E ≥ e -C e f B κ -1 L eucl Π γ| [0,τ ] \ E )
Since Π γ| [0,τ ] contains the geodesic segment joining x to ȳ,

L eucl Π γ| [0,τ ] ≥ ρ. whereas L eucl (Π(E)) ≤ L eucl (Π(∪B α )) ≤ κ α 2r(B α ) ≤ 2κǫ 3 2 ρ.
With the choice of ǫ given by ( 24), we get

L eucl (Π(E)) ≤ 1 2 ρ hence L eucl Π γ| [0,τ ] \ E ) ≥ 1 2 ρ.
Eventually using [START_REF] Grigor'yan | Stability results for Harnack inequalities[END_REF] we obtain:

L g f (γ) ≥ Ce f B rρ ≥ C (µ f (B x,y )) 1 n .
This concludes the proof of Theorem 5.7.

Let us now show that the previous proof implies that the distances d f and d 0 are α-bi-Hölder for all α ∈ (0, 1). Proposition 5.9. Let (M, g 0 ) be a closed Riemannian manifold. If f : M → R is such that df L n ≤ I Then for each α ∈ (0, 1) there is a constant C(g 0 , I, α) such that for any x, y ∈ M µ f (M )

1 n C(g 0 , I, α) d 1 α 0 (x, y) ≤ d f (x, y) ≤ C(g 0 , I, α) µ f (M ) 1 n d α 0 (x, y).
Proof. We have shown in Theorem 5.7 that the two weights e nf and e -nf are Strong A ∞ weight. In particular, there is a constant C depending on g 0 and I such that for all x, y ∈ M :

1 C d n f (x, y) ≤ µ f (B x,y ) ≤ C d n f (x, y) 1 C d n -f (x, y) ≤ µ -f (B x,y ) ≤ C d n -f (x, y)
The estimate [START_REF] Grigor'yan | Stability results for Harnack inequalities[END_REF] implies that for each λ ∈ R there is a constant C ′ depending on g 0 , λ and I such that for any x, y:

1 C ′ d n 0 (x, y) ≤ ˆBx,y e λf dµ 0 ˆBx,y e -λf dµ 0 ≤ C ′ d n 0 (x, y).
Hence, there is a constant Λ such that for any x, y ∈ M :

(25) 1 Λ d 2n 0 (x, y) ≤ d n f (x, y) d n -f (x, y) ≤ Λ d 2n 0 (x, y).
Notice that this implies that for some constant C:

1 C ≤ µ f (M )µ -f (M ) ≤ C.
But with the Hölder inequality, we get that for any p > 1 and x, y ∈ M :

d n f (x, y) ≤ Cd n 0 (x, y) Bx,y e p p-1 f dµ 0 1-1 p ,
and again the estimate [START_REF] Grigor'yan | Stability results for Harnack inequalities[END_REF] implies that Bx,y

e p p-1 f dµ 0 1-1 p ≤ C µ f (B x,y ) d n 0 (x, y) ≤ C µ f (M ) d n 0 (x, y)
.

So that we get that for any x, y ∈ M and any p > 1,

d n f (x, y) ≤ C d n p
0 (x, y)µ f (M ) which is the right-hand side inequality of Proposition 5.9. Moreover, since

d n -f (x, y) ≤ C d n p 0 (x, y)µ -f (M ) ≤ C d n p 0 (x, y) (µ f (M )) -1 ,
using the comparison (25) we get:

d 2n 0 (x, y) ≤ C d 0 (x, y) n p d f (x, y) n (µ f (M )) -1
We get therefore for all p > 1,

d 2n-n p 0 (x, y) ≤ C d n f (x, y) (µ f (M )) -1 ,
which is the left-hand side inequality of Proposition 5.9.

CONFORMAL DEFORMATIONS

OF R n WITH SCALAR CURVATURE IN L n/2
In this section, we present some interesting application of our techniques to conformal deformations of (R n , eucl).

Let us first provide the following sufficient criterion to get strong A ∞ weights.

Theorem 6.1. Let f : R n → R be such that

df L n (R n ,eucl) ≤ I
for some I > 0. Then the weight e nf is a strong A ∞ weight with respect to eucl, with constants (θ, η), where η depends only on g 0 and θ depends only on g 0 and I.

This slightly improves Theorem 3.1 of [START_REF] Bonk | The quasiconformal Jacobian problem[END_REF], since our statement is scale invariant. The proof of this result is an immediate adaptation of our proof of Theorem 5.7. We will not provide further details since it would be a mere repetition of the previous section.

Nevertheless, we show now that Theorem 6.1 implies Theorem 1.6, which provides a completely new criterion to get Strong A ∞ weights on (R n , eucl).

Proof of Theorem 1.6. Let g = e 2f eucl be a conformal metric on the Euclidean space

R n>2 such that i) Vol(R n , g) = +∞; ii) ´Rn | Scal g | n 2 (x
) dvol g (x) < +∞. We will show that e nf is a strong A ∞ weight with respect to eucl.

The following lemma shows that our infinite volume assumption is often satisfied.

h(y).

Hence if h : M → R is bounded from below, setting λ = inf y∈M h(y), we get that for every geodesic ball

sup y∈B(x,r) (h(y) -λ) ≤ γ inf y∈B(x,r) (h(y) -λ).
But as r → +∞, we have lim r→+∞ inf y∈B(x,r) h(y) = λ therefore for all y ∈ M , h(y) = λ.

As an example that will be useful later, let us consider the case where ḡ = e 2u eucl is a conformal metric on the Euclidean space R n and assume that ˆRn |du| n (x)dx < +∞.

It follows from Theorem 6.1 that e nu is a strong A ∞ weight and that the metric ḡ satisfies the Poincaré inequality and the doubling condition: there are constants θ, λ such that for all ḡ-geodesic balls B(x, r), we have

Vol ḡ B(x, 2r) ≤ θ Vol ḡ B(x, r) and, writing ϕ B(x,r) := ffl B(x,r) ϕ dvol ḡ , ∀ϕ ∈ C 1 (B(x, 2r)) : ϕ -ϕ B(x,r) 2 L 2 (B(x,r),ḡ) ≤ ∇ϕ 2 L 2 (B(x, 2r
),ḡ) . Moreover, by the results of Section 4.1, the metric space (R n , ḡ, dvol ḡ ) is Alhfors regular: there is a constant η such that for every ḡ-geodesic ball B(x, r) :

ηr n ≤ Vol ḡ B(x, r) ≤ r n /η.
In particular by result in [START_REF] Grigor'yan | The heat equation on noncompact Riemannian manifolds (Russian)[END_REF][START_REF] Saloff-Coste | A note on Poincaré, Sobolev and Harnack inequalities[END_REF], ḡ satisfies the parabolic and the elliptic Harnack inequalities and if n ≥ 3, then (R n , ḡ) is a non parabolic manifold and has a positive Green kernel G ḡ (x, y) that satisfies the estimates Scal g e 2f φ.

Hence if we define ψ = φe -u we get ∆ψdu, dψ = n -2 4(n -1)

Scal g e 2f 1 B(R) ψ.

As du ∈ L n , the metric ḡ = e Scal g e 2f 1 B(R) e -4 n-2 u ψ.

As ∆ ḡψ has compact support and is in L p for every 1 ≤ p < +∞, we can find a solution of the equation

∆ ḡ ξ = ∆ ḡ ψ,
for instance let us choose ξ(x) = ˆRn G ḡ (x, y)∆ ḡ ψ(y)dvol ḡ (y).

Since supp ∆ ḡψ ⊂ K, it follows from ( 26) that there exists C > 0 such that for all x ∈ M \K, ξ(x) ≤ C ˆK G q ḡ (x, y)dvol ḡ (y)

1 q
≤ Cd ḡ (x, K) 2-n .

In particular, ´Rn |ξ(y)| 2n n-2 dvol ḡ(y) < +∞, the map ξ tends to zero at infinity and is in W 2,p loc (R n ) for every 1 ≤ p < +∞. Therefore ξ is a bounded function and ψξ is a ḡ-harmonic function that is bounded from below: it is hence a constant function. If this constant is 0 then ψ = ξ and This implies that the metric g = φ 4 n-2 eucl and ḡ = e 2u eucl are bi-Lipischitz, hence since e nu is a strong A ∞ weight with respect to g eucl , e nf = φ 2n n-2 is also.

We emphasize again on the fact that this implies very strong geometric constraints on conformal deformations of (R n , g eucl ) with

GROMOV-HAUSDORFF LIMIT AND CONFORMAL DISTANCES

Let us conclude by presenting a natural question about the limiting distances which arise in our Theorem 1.3, which we could not answer.

Let us first recall that to any strong A ∞ weight e nf∞ (with respect to dvol g0 ) is naturally associated a distance d f , defined as follows. For all x, y ∈ M , we write agin B xy the largest g 0 -geodesic ball whose diameter is the g 0 -geodesic segment It can be easily shown that e nf ∈ L 1 (M ) is a strong A ∞ weight if and only if d f is a distance (see [START_REF] Semmes | Bi-Lipschitz mappings and strong A∞ weights[END_REF] for more details). Moreover, an elementary approximation argument shows that if f is continuous, then d f coincides with the classical Riemannian distance associated to the Riemannian metric g f = e 2f g 0 .

Let us now consider a sequence (g k = e 2f k g 0 ) k∈N of conformal metrics satisfying the hypotheses of Corollary 1.4. Up to extracting a subsequence, the sequence of associated metric space (M, d g k ) converges in Gromov-Haudsorff topology to some distance d ∞ on M , which is bi-Hölder equivalent to d 0 . Moreover, still up to extracting a subsequence, the Riemannian volume measure µ k = dvol g k = e nf k dvol g0 converge in the weak- * topology to a positive measure µ ∞ on M . Since all the µ k are uniformly strong A ∞ weights with respects to dvol g0 , µ ∞ is also: there exists hence e nf∞ ∈ L 1 (M ) which is still a strong A ∞ weight with respect to g 0 . This weight is therefore associated to a distance d f∞ defined by [START_REF] Ladyzhenskaya | Estimates for the Hölder constant for functions satisfying a uniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients[END_REF]. Does d ∞ and d f∞ coincide ?

The reader will can check that in the Example 2 presented in our introduction, where the conformal factors converge in W 2, n 2 to the singular conformal factor given by 1) and the distances converge in the Gromov-Hausdorff topology, these distances indeed coincide.

One can easily show from semi-continuity arguments that d ∞ ≤ d f∞ . Let us present now a striking example of pathological behaviour which shows that in general, for a sequence of (unifom) strong A ∞ weights (e nf k ) k∈N , the converse inequality is not true. The following examples of conformal deformations is due to D. Burago, in [START_REF] Burago | Periodic metrics, Representation theory and dynamical systems[END_REF] Let us consider the standard flat torus (T n = R n /Γ, eucl), where Γ = (2πZ) n . For all ℓ > 1, we consider the conformal deformation of the euclidean metric g 0 given by g ℓ = e 2f ℓ eucl, where e nf ℓ (x 1 , . . . , x n ) = 1 -1 2 cos(ℓx 1 ).

Since e nf ℓ is uniformly bounded from above and below, all the metric space (T n , d ℓ = d f ℓ ) are uniformly bilipschitz to T n equiped with the Euclidean distance and the (e nf ℓ ) ℓ≥1 are uniformly strong A ∞ weights with respect to the Lebesgue metric. Nevertheless, the distance associated to the limit of the volume measures and the limit of distances are radically distinct as shows the following result.

Theorem 7.1. Under the above notations, as ℓ → +∞,

• the sequence (e nf ℓ ) ℓ converges weakly in L 1 (T) to the function 1;

• the sequence of metric spaces (T n , d ℓ ), ℓ ∈ N converges in the Gromov-Hausdorff topology to metric space (T n , d ∞ ) where d ∞ is the Finsler metric associated to the stable norm of the periodic metric on R n given by ḡ = 1 -1 2 cos(x 1 )

2 n
eucl.

Proof. The first assertion is classical. As for the second assertion, let us recall that if d is the Riemannian distance on R n associated to the metric ḡ then the associated stable norm • * is defined for all x ∈ R n by x * = lim t→+∞ Hence the result.

4 n

 4 -2 g 0 , i.e. e f = u 2 n-2 . The proof is done in two steps. Lemma 3.8. Under the same hypothesis of Theorem 3.7, there exist ε

  John-Nirenberg inequality (cf [20, Theorem 7.21]), there exists ε > 0 such that for any geodesic ball B of radius less that R 0 /2, B e ε|ln u-(ln u) B | dµ 0 ≤ C, where (ln u) B = B ln(u) dµ 0 . This implies that B u ±ε dµ 0 ≤ Ce ±ε(ln u) B .

4. 1 . 1 .

 11 Definitions.

n 2 (

 2 B(x0,r0)) ≤ βc then λ 0 (c) ≥ 0. For instance for c + = 2 β q L n 2 , we have λ 0 (c + ) ≥ 0. Hence there is a c 0 ∈ [c -, c + ] such that λ 0 (c 0 ) = 0.

n 2 ≤

 2 C(g 0 )I. We also have log ϕ = f + w and ∆ log ϕ -|d log ϕ| 2 = V Hence by Stokes formula,

Be

  λf dµ 0 ≤ e λf B e ( |λ| c df L n (B)) n B exp γ |f (x)f B | df L n (B) n n-1 dµ 0 (x) ≤ C(g 0 , |λ|, I)e λf BUsing the Cauchy-Schwarz inequality, this implies1 ≤ B e λf dµ 0 × B e -λf dµ 0 ≤ C(g 0 , |λ|, I) B e ±λf dµ 0 e λf B ,and we eventually get (22) e λf B C(g 0 , |λ|, I) ≤ B e λf dµ 0 ≤ C(g 0 , |λ|, I)e λf B .

  r) := ´B(x,r) |df (y)|dµ 0 (y) and one gets |f (x)f B | ≤ ˆB(x,2r(B)) C d n-1 0 (x, y) |df (y)|dµ 0 (y) ≤ C (2r(B)) n-1 V (x, 2r(B)) + ˆ2r(B) 0 (n -1)C r n V (x, r)dr

  Let now x, y ∈ M such that ρ := d 0 (x, y) ≤ 10η and let γ : [0, 1] → M be a C 1 curve joining x to y: γ(0) = x and γ(1) = y Let τ > 0 be the first time with d 0 (x, γ(τ )) = ρ and let ȳ = γ(τ ), B := B x,ȳ = B(m, ρ/2). We have for any t ∈ [0, τ ], γ(t) ∈ 3B. Let c(t) = exp m (tu) be the unit speed g 0 geodesic joining x to ȳ and L := c([-3ρ/2, 3ρ/2]) it is a diameter of 3B and Π : 3B → L the projection onto L associated to the Euclidean metric d eucl induced by exp m . We know that Π is 1-Lipschitz for the Euclidean distance and for x, y ∈ L : d eucl (x, y) = d 0 (x, y). Hence Π is κ-Lipschitz for the distance d 0 . According to Lemma 5

2 → 2 , n 2 loc

 222 y) ≤ G ḡ (x, y) ≤ Cd 2-n ḡ (x, y) or equivalently :c (Vol ḡ(B x,y ))) -1+ 2 n ≤ G ḡ(x, y) ≤ C (Vol ḡ (B x,y ))) -1+ 2 n .Let us go back to our metric g = e 2f eucl withVol(R n , g) = +∞ and ˆRn | Scal g | n 2 (x) dvol g (x) < +∞.Let ∆ be the Euclidean Laplacian and let A n be the norm of the operator d∆ -1 : L n L n . Since the L n/2 norm of Scal g is finite, for R large enough we haven -2 4(n -1) ˆRn \B(R) | Scal g | n 2 (x) dvol g (x)was done in the proof of Proposition 5.4, we can findv ∈ L n 2 (R n , dx) such that v -d∆ -1 v 2 = n -2 4(n -1) Scal g e 2f 1 R n \B(R) . Now, let u ∈ W (R n ) be such that ∆u = v, for instance u is a primitive of d∆ -1 v ∈ L n or u(x) = c n ˆRn 1 xy n-2 -1 y n-2 v(y)dy.We have ∆e u = n -2 4(n -1)Scal g e 2f 1 R n \B(R) e u .Writing φ = e

4 n- 2 u

 42 eucl induces a strong A ∞ weight, but∆ ḡ ψ = e -4 n-2 u (∆ψdu, dψ ) = n -2 4(n -1)

  Vol(R n , g) = ˆRn ψ(y) 2n n-2 dvol ḡ (y) = ˆRn ξ 2n n-2 (y)dy < +∞.This is contradictory with our hypothesis hence there is a positive constant c such thatψ = c + φ,hence there are positive constants α, A such that α ≤ ψ ≤ A.

ˆRn | Scal g | n 2

 2 (x) dvol g (x) < +∞ and Vol g (R n , g) = +∞. It implies in particular Poincaré inequality, Euclidean-type isoperimetric inequality and Sobolev inequalities, which were unknown on such examples.

  [xy]. Let us writeδ f (x, y) = 1 ω n µ f (B xy ) e nf (x) dvol g0 (x) 1/n, where ω n is the Euclidean volume of the unit n-ball, and define(27) d f (x, y) = lim ε→0 inf N i=0 δ f (x i , x i+1 ) ; x 0 = x, x N +1 = y, d 0 (x i , x i+1 ) ≤ ε .

1 t

 1 d(0, tx). Using the change of variableξ i = ℓx i , i = 1 . . . n, we get that (T n , d ℓ = d f ℓ ) is isometric to the quotient R n /(ℓΓ) with the metric 1 ℓ 2 ḡ. More exactly let π : R n → T n = R n /Γ the quotient map. if x, y ∈ R n then d ℓ (π(x), π(y)) = inf γ∈Γ 1 ℓ d(x, y + ℓγ) If d ∞ (π(x), π(y)) = inf γ∈ℓΓ xyγ * ,then we clearly have sup x,y∈R n |d ℓ (π(x), π(y))d ∞ (π(x), π(y))| ≤ C ℓ .

  Note that on the round sphere, we have ˆSn |Q gS | dvol gS = (n -1)!σ n :

2,n

with bounds only depending on C, δ, n and R 0 .

  Now, our proof of Theorem 1.6 will be based on the following elementary lemma.

	∀ϕ ∈ C 1 0 (R n ) : ˆ|ϕ|	n-2 dvol g 2n	1-2
	Lemma 6.3. Let (M, g) be a complete Riemannian manifold that satisfies the elliptic Har-
	nack inequality. Then harmonic functions which are bounded from below are constant.
	Proof. Harnack inequality establishes the existence of a constant γ > 0 such that for every
	geodesic ball B(x, r) ⊂ M and every positive harmonic function h defined over B(x, 2r)
	one has
	sup y∈B(x,r)	h(y) ≤ γ inf y∈B(x,r)

Lemma 6.2. A complete conformal deformation g = e 2f eucl of the Euclidean metric on R n>2 that satisfies ´Rn | Scal g | n 2 (x) dvol g (x) < +∞ has necessary infinite volume. Proof. According to this hypothesis, the Riemannian manifold (R n , g) satisfies the Sobolev inequality ([12, Proposition 2.3] ): for some C > 0, n ≤ C ˆ|dϕ| 2 g dvol g .

Hence (R n , g) has an infinite volume, since the volume of geodesic balls is then bounded from below (

[START_REF] Akutagawa | Yamabe metrics of positive scalar curvature and conformally flat manifolds[END_REF][START_REF] Carron | Inégalités isopérimétriques de FaberKrahn et conséquences[END_REF]

): c(n, µ)r n ≤ Vol g (B(x, r)).

We emphasize on the fact that in this example, d f∞ = d eucl , whereas d ∞ is a Finsler non-Riemannian metric: it cannot in any weak sense be considered as a conformal metric for some limiting conformal factor. The reader will easily notice than in the previous example, the volume of the conformal metrics are uniformly bounded, but the L n/2 -norm of the scalar curvature blows up. Under the L n/2 assumptions on the scalar curvature which ensures Theorem 1.3 and Corollary 1.4, we have a stronger control on the geometrical and analytical behaviour of the conformal factors than the information given by the strong A ∞ bounds. Nevertheless, we cannot show yet equality for d ∞ and d f∞ under these hypotheses, nor give a counterexample.

Moreover, by Hölder inequality we have

Therefore

, which concludes the proof of Lemma 5.6.

We can now prove Theorem 5.1.

Proof of Theorem 5.1. We will use the constants A(g 0 ) and β given by the estimate [START_REF] Fefferman | Q-curvature and Poincar metrics[END_REF] and the Sobolev inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. We can always assume that ρ also satisfies

.

We choose a finite cover

then for any i ∈ {1, . . . , N }, using Lemma 5.6, we find c i such that with

the lowest eigenvalue of the Schrödinger operator ∆q i is zero. We have

Hence using Proposition 5.4, we know that if

then we find ϕ i ∈ ker(∆q i ), ϕ > 0, with

On the other hand, ∆ log

We choose ǫ such that 24

Volg 0 (M)

Let ϕ : M → (0, +∞) be a global solution of ∆ϕ = V ϕ. For all i ∈ {1, ..., N }, we define ψ i : B(x i , ρ) → R by ϕ = ϕ i ψ i .

Since on the ball B(x i , ρ), we also have ∆ϕ i = V ϕ i , we get ∆ψ i -2 d log ϕ i , dψ i = 0.