Samuel Bronstein

ON THE MOSER-TRUDINGER INEQUALITY FOR SURFACES

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Introduction

In this paper we consider (Σ, σ) a Riemannian surface, and we look at the famous Sobolev embeddings W 1,2 (Σ) ֒→ L p (Σ) for any p ≥ 2.

When Σ is of infinite volume, we prove the following version of the celebrated Moser-Trudinger inequality: Theorem 1.1. Let (Σ, σ) be an infinite volume complete surface. Assume that the curvature of Σ is upper bounded by K. Assume that the systole δ and the Cheeger isoperimetric constant h are both nonzero. Then there is a constant C depending only on (δ, h, K) such that, for any u ∈ W 1,2 (Σ):

(1.1)

Σ |∇u| 2 dσ ≤ 1 ⇒ Σ exp 4πu 2 -1dσ ≤ C(δ, h, K)
We refer to section 2 for precise definitions of systole and isoperimetric constant for noncompact surfaces.

For compact surfaces, this statement cannot be true because of the existence of nonzero harmonic maps. The standard way to deal with this problem is to add the assumption that our maps have zero average, restricting effectively to the orthogonal of harmonic maps in W 1,2 .

In this setting, we prove the following inequality:

Date: 2023.
Theorem 1.2. Let a > 0 and K ∈ R. There is a constant C(a, δ, h, K) such that, for any closed surface (Σ, σ) with systole δ, Cheeger constant h, curvature upper bounded by K and volume bigger than a, for any u ∈ W 1,2 (Σ) satisfying:

(1.2)

Σ udσ = 0, Σ |∇u| 2 dσ ≤ 1
Then:

(1.3) Σ exp(4πu 2 ) -1 dσ ≤ C(a, δ, h, K)

When the considered surface is the sphere S 2 with its round metric, our theorem is exactly the Moser-Trudinger inequality, first proved by Trudinger [START_REF] Trudinger | On imbeddings into orlicz spaces and some applications[END_REF] and then generalized by Moser to the higher-dimensional spheres [START_REF] Moser | A sharp form of an inequality by n. trudinger[END_REF].

When considering the hyperbolic plane H 2 which satisfies the condition of the abovementioned theorem, we get back a result of Mancini and Sandeep [START_REF] Mancini | Moser-trudinger inequality on conformal discs[END_REF].

The constant 4π is the biggest one for which the statement is true, statement proven by Li [START_REF] Li | Moser-trudinger inequality on compact riemannian manifolds of dimension two[END_REF] for any compact Riemannian surface.

We insist that the main novelty is the dependency of the constant C involved in the parameters, and particularly that it is roughly independent of the volume of the surface. There are other ways to prove the Trudinger-Moser inequality for compact surfaces, by a concentration argument, as Li did [START_REF] Li | Moser-trudinger inequality on compact riemannian manifolds of dimension two[END_REF]. We would expect other more recent methods to work to, for instance the fractional Sobolev inequalities have recently been used to prove an analogous inequality in the case of the n-dimensional sphere , see Xiong [START_REF] Xiong | A derivation of the sharp moser-trudinger-onofri inequalities from the fractional sobolev inequalities[END_REF].

The same kind of dependency is also true for the constants involved in the Sobolev embeddings, see Ilias's work [START_REF] Ilias | Constantes explicites pour les inégalités de sobolev sur les variétés riemanniennes compactes[END_REF]. See also Faget [Fag06], for a generalization of the exponential decay of Sobolev's constants in higher dimensional manifolds.

This reinforces the feeling that Moser-Trudinger is a nice way of understanding the asymptotic behavior of the embeddings W 1,2 (Σ) ֒→ L p (Σ).

Generally, we have no idea of the sharp constant C involved in the theorem. This question was asked in a paper of Dolbeault-Esteban-Jankowiak [START_REF] Dolbeault | Onofri inequalities and rigidity results[END_REF]. Our paper gives a partial answer as it claims that the constant won't explode if we control the right parameters, but we think the exhibition of the sharp constant should involve more complicated techniques.

Our proof mainly relies on a symmetrisation argument, which naturally involves the isoperimetric inequality. That's why we preferred to write the dependency in terms of the isoperimetric constant rather than the spectral gap, even if they are closely linked by the Cheeger and Buser inequalities [START_REF] Cheeger | A lower bound for the smallest eigenvalue of the laplacian[END_REF][START_REF] Buser | A note on the isoperimetric constant[END_REF].

Trudinger-Moser inequality has had tremendous consequences regarding the prescribed curvature problem on S 2 . Famously, Moser used it to solve the problem of prescribed curvature on the projective plane [START_REF] Moser | On a nonlinear problem in differential geometry[END_REF]. use their version of Moser-Trudinger for the prescribed curvature problemn on subsets of R 2 . It has also been used by Jost-Wang [START_REF] Jost | Analytic aspects of the toda system: I. a moser-trudinger inequality[END_REF] to solve some Toda systems on closed surfaces. Malchiodi-Ruiz [START_REF] Malchiodi | A variational analysis of the toda system on compact surfaces[END_REF] and [START_REF] Battaglia | A general existence result for the toda system on compact surfaces[END_REF] deduced existence theorems for mean-field equations from the Trudinger-Moser inequality.

Our focus here is rather on hyperbolic surfaces of high genus. A corollary of our theorem is the possibility of taking high degree covers of a fixed hyperbolic surface keeping a uniform sobolev embedding Corollary 1.3. Let Σ be a closed hyperbolic surface. There are sequence (Σ n ) of Riemannian covers of Σ of degree n and C(Σ) > 0 such that, for any n ≥ 1, u ∈ W 1,2 (Σ n ) satisfying

(1.4) Σn u = 0, Σn |∇u| 2 ≤ 1 then (1.5) Σn e 4πu 2 -1 ≤ C(Σ)
Indeed, thanks to Magee-Naud-Puder [START_REF] Magee | A random cover of a compact hyperbolic surface has relative spectral gap 3 16-ε[END_REF] we know that for any ε > 0, with probability going to one when the index of the covering increases, a random cover of X has relative spectral gap bigger than 3 16ε. This means that there are sequences of coverings of a closed surface with spectral gap uniformly lower bounded, hence for such a sequence (X n ) the Cheeger constant is lower bounded. The systole and volume are bigger than the systole and volume of X, and the curvature is -1, hence the constant in Moser-Trudinger is uniform.

The author is very thankful to Nicolas Tholozan and Paul Laurain for their time and discussion on these topics.

Notations and isoperimetric comparison

Throughout this section (Σ, σ) will be a Riemannian surface. Recall the definitions of systole and Cheeger constant: Definition 2.1 (Systole). The systole of (Σ, σ) is the length of the shortest noncontractible closed curve in Σ.

Definition 2.2 (Cheeger constant). Denote l the length measure associated to σ. The Cheeger constant of (Σ, σ) is:

(2.1) h := inf A⊂Σ, 2Vol(A)<Vol(M ) l(∂A) Vol(A)
If Vol(Σ) = ∞, we require A to be compact.

The Cheeger constant is heavily linked with the spectral gap of the surface.

Definition 2.3. For a closed surface, the spectral gap Λ of (Σ, σ) is the smallest nonzero eigenvalue of -∆.

It is a standard fact that the spectrum of the Laplacian on a compact surface is discrete, hence the spectral gap of a closed surface is always strictly positive. Remark that there is a way to generalize the spectral gap for asymptotically negatively curved surfaces, using the resolvent of the laplacian. See for instance Borthwick [START_REF] Borthwick | Spectral theory of infinite-area hyperbolic surfaces[END_REF] for the spectral gap of infinite area hyperbolic surfaces. On the hyperbolic plane, the spectral gap will be 1 4 , which is a reformulation of Hardy's inequality. The spectral gap and Cheeger's isoperimetric constant are closely linkedn thanks to two relations. Recall the first one, proved by Cheeger in [Che70]:

Theorem 2.4 (Cheeger). Let Σ be a complete surface. Denote h and Λ its Cheeger constant and spectral gap. Then

(2.2) 4Λ ≥ h 2 .
Remark that Cheeger states it for closed surfaces, but it has since been extended for complete noncompact surfaces, see [BPP + 93]. Buser [START_REF] Buser | A note on the isoperimetric constant[END_REF] , theorem 7.1., showed that for surfaces with curvature bounded below, a converse holds: Theorem 2.5 (Buser). There is a universal constant C, such that: Let (Σ, σ) be a complete Riemannian surface with curvature bounded below by -δ 2 , δ ≥ 0. Denote h and Λ its Cheeger constant and spectral gap. Then

(2.3) Λ ≤ C(δh + h 2 )
We will need a stronger insight on the isoperimetric behavior of a surface, hence we introduce the isoperimetric profile, as considered by Grimaldi-Nardulli-Pansu [GNP09]: Definition 2.6 (Isoperimetric profile). Let (Σ, σ) be a Riemannian surface. The isoperimetric profile is the following map:

(2.4) ∀t > 0, ϕ(t) = inf A⊂Σ,Vol(A)=t l(∂A)
Remark that this notation differs from the isoperimetric profile for revolution manifolds considered by Gallot in [START_REF] Gallot | Inégalités isopérimétriques et analytiques sur les variétés riemanniennes[END_REF], but it is easy to go from one definition to another.

Lemma 2.7. Let Σ be a complete surface with nonzero systole δ and curvature bounded above by K. Denote ϕ its isoperimetric profile. Denote V 0 the following:

(2.5) V 0 = δ 2 2π + √ 4π 2 -Kδ 2 Consider t 0 = min{V 0 , Vol(Σ) 2 } if K ≤ 0 and t 0 = min{V 0 , Vol(Σ) 2 , 4π 1+K } if K > 0
Then for t ≤ t 0 , we have:

(2.6) ϕ(t) ≥ t(4π -Kt) Proof. Assume first V 0 ≤ Vol(Σ) 2
Let A ⊂ Σ be a domain of area t ≤ V 0 . V 0 was chosen so that:

(2.7) V 0 (4π -KV 0 ) ≤ δ
Hence the wanted inequality is obviously verified if l(∂A) ≥ δ.

Otherwise, if l(∂A) ≤ δ, because of completeness one can assume that A is a disjoint union of disks. But for a disk, as the curvature is lower than K, the length of its boundary is bigger than the length of a disk of same area in constant curvature K model space. Hence, we get the inequality desired.

We will compare the isoperimetric profile of a surface with the behavior of a radial metric on the disk.

Lemma 2.8. Consider ε > 0, h > 0 and K ∈ R. Let g be defined as the following map:

(2.8) ∀t > 0, g(t) :=    t(4π -Kt) if t ≤ ε g(ε) if ε < t ≤ g(ε) h h • t if t > g(ε)
h

Then there exists a C 1 map f : R + → R + such that the disk endowed with radial metric dr 2 + f ′ (r) 2 dθ 2 satisfies:

(1) The perimeter of a centered disk of radius A is g(A)

(2) there is a constant C(g) > 0 such that the metric g D = dr 2 + f ′ (r) 2 dθ 2 is uniformly conformal to a disk with constant negative curvature. Proof.

The assumption (1) is verified if and only if f satisfies the functional equation:

(2.9) 2πf ′ (r) = g(2πf (r)) ∀r > 0
This equation can be explicitly solved on (0, ε) with:

(2.10)

f (r) =      -2 K sinh( √ -Kr 2 ) if K < 0 r 2 2 if K = 0 2 K sin( √ Kr 2 ) if K > 0
On (ε, ∞) there is a nondecreasing solution on R, which has at least linear growth, as g is lower bounded. The solution is defined on all of R because g is sublinear. Actually, we have the following explicit description of f :

∀ε < t ≤ g(ε) h , f (t) = f (ε) + (t -ε) g(ε) 2π (2.11) ∀t > g(ε) h , f (t) = f ( g(ε) h ) exp h(t - g(ε) h ) (2.12) As for t > g(ε)
h , we have:

(2.13) f ′ (t) sinh(ht) = 2hf ( g(ε) h )e -g(ε) 1 1 -e -2ht
And f ′ (t) sinh(ht) has a nonzero limit when t → 0. We deduce that there is a constants C(K, ε, h) so that

(2.14) 1 C(K, ε, h) ≤ f ′ (t) sinh(ht) ≤ C(K, ε, h)
so f ′ is uniformly comparable to sinh(hr), which means that our disk is uniformly quasiconformal to the complete disk with constant sectional curvature -h. As this disk is quasiconformal to the hyperbolic plane, we get the statement (2). We point out that the quasiconformal factor depends only on the map g, so in the parameters (K, ε, h).

The Moser-Trudinger inequality for infinite-volume surfaces

It is known that a Moser-Trudinger type inequality can't be true for all noncompact surfaces, as it implies the existence of a Poincaré inequality. We will show here that for a class of surfaces which at infinity, look like they are negatively curved, a Moser-Trudinger inequality is true. We will also show that the relevant geometric data are the systole and Cheeger constant. Throughout this section (Σ, σ) will be a Riemannian surface, noncompact. We will show that the following geometric assumptions are enough to get a Moser-Trudinger inequality:

Theorem 3.1. Let (Σ, σ) be an infinite volume complete surface. Assume that the curvature of Σ is upper bounded by K. Assume that the systole δ and the Cheeger isoperimetric constant h are both nonzero. Then there is a constant C depending only on δ, h, K such that, for any u ∈ W 1,2 (Σ):

(3.1) Σ |∇u| 2 dσ ≤ 1 ⇒ Σ exp 4πu 2 -1dσ ≤ C(δ, h, K)
Note that the Moser-Trudinger with exponent 4π was already shown by Mancini-Sandeep [START_REF] Mancini | Moser-trudinger inequality on conformal discs[END_REF] in the case of disks dominated by the hyperbolic metric. Our proof relies on a radial rearrangement adapted to the surface Σ, the same used by Ilias [START_REF] Ilias | Constantes explicites pour les inégalités de sobolev sur les variétés riemanniennes compactes[END_REF] to estimate the constants involved in the Sobolev embeddings. The following formulas are well-known measure-theoretic tools: Lemma 3.2. Let u be a positive C 2 Morse function. Assume it is W 1,2 (Σ). Denote ν the length measure associated to σ. Consider the functions l(t) and A(t):

A(t) := σ{u ≥ t} (3.2) l(t) := ν{u = t} (3.3)
Then, for any measurable function f : R → R :

Σ f (u) = R f (t)A(t)dt (3.4) Σ |∇u| 2 ≥ R l 2 (t) A ′ (t) dt (3.5)
Proof. The first term is a standard formula in differential geometry, it is proven via a change of variables. For the second inequality, we use the following formula for the derivative of A:

(3.6) A ′ (t) = {u=t} dν |∇u|
Now the co-area formula yields (see [START_REF] Srinivasan Kesavan | Symmetrization and applications[END_REF], theorem 2.2.2)

(3.7) Σ |∇u| 2 = R {u=s} |∇u|dν ds
And finally, applying Cauchy-Schwarz:

(3.8)

{u=s} |∇u|dν {u=s} dν |∇u| ≥ l 2 (s)
which finishes the proof.

Theorem 3.3. Let (Σ, σ) be an infinite volume complete surface, with nonzero systole, Cheeger constant and upper bounded curvature δ, h, K. There is an nondecreasing function f depending only on (δ, h, K), a disk uniformly conformal to the complete hyperbolic plane, with radial metric dr 2 + f ′ (r) 2 dθ 2 . Moreover, the radial rearrangement mapping on that disk:

(3.9) W 1,2 (Σ) → W 1,2 (D) u → u *
defined so that u * is nonincreasing, radial, and:

(3.10)

A(t) = |{u * ≥ t}| ∀t ∈ R satisfies: Σ f (u)dσ = D f (u * ) (3.11) Σ |∇u| 2 ≥ D |∇u * | 2 (3.12)
Proof. Denote ϕ the isoperimetric profile of Σ, σ By lemma 2.7, there is V 0 (K, δ) > 0 such that, on (0, V 0 ) the isoperimetric profile satisfies:

(3.13) ϕ(t) ≥ t(4π -Kt)
Consider the map g defined in lemma 2.8 with ε = V 0 . Then we claim:

(3.14) ∀t ∈ R, ϕ(t) ≥ g(t)
Indeed, for small values it is the lemmma 2.7, for big values it is the definition of the Cheeger constant and in between it comes from the fact that g(V 0 ) ≤ δ.

Hence define f by the lemma 2.8. For u C 2 , Morse, nonnegative and W 1,2 on Σ define u * its radial rearrangement on D, dr 2 + f ′ (r) 2 dθ 2 . By construction, it is clear that

(3.15) Σ f (u) = D f (u * )
for any measurable f . Also, denote l and l * the length of level set functions:

l(t) = ν{u = t}| (3.16) l * (t) = |{u * = t} (3.17)
Because g ≤ ϕ everywhere, it is clear that l ≥ l * . Thanks to lemma 3.2, we deduce

(3.18) Σ |∇u| 2 ≥ D |∇u * | 2
Proof of theorem 3.1. Consider Σ such a surface. Consider u ∈ W 1,2 (Σ). First we will prove the inequality for Morse surfaces, then by density it will be true on W 1,2 (Σ).

As the functional e 4πx 2 is even and the absolute-value is 1-lipschitz, up to replacing u by |u|, we can assume that u is nonnegative and

(3.19) Σ |∇u| 2 ≤ 1 .
Consider (D, dr 2 +f ′ (r) 2 dθ 2 ) the disk defined in theorem 3.3 and u * the radial rearrangement. By construction,

D exp 4π(u * ) 2 -1 = Σ exp 4πu 2 -1 (3.20) D |∇u * | 2 ≤ 1 (3.21)
As D |∇u * | 2 is a conformal invariant, consider C the conformal distance between dr 2 + f ′ (r) 2 dθ 2 and the metric of the Poincaré disk. Naturally, with C being the conformal distortion, we control: [START_REF] Mancini | Moser-trudinger inequality on conformal discs[END_REF], this is upperbounded by some bound depending only on h. Finally, as C depends only on (δ, h, K) we get a bound depending on those parameters. We finish our discussion about infinite volume surfaces with 3 examples of noncompact surfaces which do not satisfy the Moser-Trudinger inequality, showing that none of the 3 assumptions could have been dropped. Proof. The main tool for this is to show that the constant of Moser-Trudinger explodes in high positive curvature. Fix M > 0 Let Ω be a small disk on which κ ≥ M Consider C S the Moser-Trudinger constant for the sphere. Thanks to works of Chang and Yang, there is a bubbling sequence u n of functions on S 2 , weakly converging towards a Dirac distribution, so that:

(3.22) ∞ 0 (exp 4π(u * ) 2 -1)f ′ (r)dr ≤ C ∞ 0 (exp 4π(u * ) 2 -1) sinh(r)dr Now, cf
S 2 |∇u n | 2 ≤ 1 (3.24) S 2 exp 4πu 2 n ) -1 → C S (3.25)
Because of the weak convergence, one can assume that Suppu n is in small neighborhood of a point. Hence for any ε > 0 we easily get a function v supported in Ω satisfying:

Σ |∇v| 2 ≤ 1 (3.26) Σ exp(4πv 2 ) -1 ≥ MC S -ε (3.

A Moser-Trudinger inequality for closed surfaces

In this section, we need to adapt our proof with more subtlety in order to get the same theorem for closed surface. We point out that Li [START_REF] Li | Moser-trudinger inequality on compact riemannian manifolds of dimension two[END_REF] already proved that all compact surfaces satisfy the Moser-Trudinger inequality with exponent 4π, however his proof does not show the dependency of the upper bound in the systole, curvature and Cheeger constant.

Theorem 4.1. Let Σ, σ be a closed surface with nonzero systole δ, Cheeger constant h and upper bounded curvature K. Then there is a constant C(δ, h, K, Vol(Σ)) such that, for any u ∈ W 1,2 (Σ) with zero average and satisfying:

(4.1) Σ |∇u| 2 dσ ≤ 1 Then (4.2) Σ exp 4πu 2 -1 dσ ≤ C(δ, h, K) .
The constant C can be chosen to be nonincreasing in Vol(Σ). Remark 4.1. In the case of hyperbolic surfaces, an important feature is that the bound is actually independent of the volume, as long as the systole and spectral gap are controlled.

In the case of tori, it actually depends on the volume, as big volume flat tori must have small Cheeger constant.

When dealing with zero average function, we cannot assume, of course, that they are nonnegative. Hence we must be more careful when considering radial rearrangmeents: Theorem 4.2. Let (Σ, σ) be a closed surface. Debite δ, h, K its systole, Cheeger constant and supremum of curvature. Then there is a disk of radius R = Vol(Σ) 2 , with radial metric dr 2 + f ′ (r) 2 dθ 2 , so that for any u Morse function on Σ one can define u -and u + its lower and upper radial rearrangements satisfying:

(1) u + is nonincreasing radial on D.

(2) u -is nondecreasing radial on D (3)

∀t such that σ(u ≥ t) < Vol(Σ) 2 , σ(u ≥ t) = |{u + ≥ t}| (4.3) ∀t such that σ(u ≤ t) < Vol(Σ) 2 , σ(u ≤ t) = |{u -≤ t}| (4.4) (4) For any measurable f , Σ f (u)dσ = D f (u -) + f (u + ) (4.5) Σ |∇u| -2 2 ≥ |∇u + | 2 2 + |∇u -| 2 2 (4.6)
Proof. Denote ϕ the isoperimetric profile of Σ. Thanks to lemma 2.7, there is V 0 > 0 so that, on (0, V 0 ):

(4.7) ϕ(t) ≥ t(4π -Kt)
Also, on (0, Vol(Σ) 2 , by definition of the Cheeger constant: (4.8)

ϕ(t) ≥ h • t
Consider the map g defined in lemma 2.8 with ε = V 0 . We check

(4.9) ∀t ≤ Vol(Σ) 2 , ϕ(t) ≥ g(t)
Hence define f by the lemma 2.8, and consider u + the nonincreasing radial rearrangement of the part of u above its median m. In the same way, define mu -to be the nonincreasing radial rearrangement of the part of mu above u. Thanks to the formulas 3.2, we deduce easily, for any measurable f :

(4.10) Σ f (u) = D f (u + ) + f (u -)
And because g is lower than the isoperimetric profile on (0, Vol(Σ) 2 ), we deduce:

(4.11) Σ |∇u| 2 ≥ |∇u + | 2 2 + |∇u -| 2
As a direct consequence, one can mimic the proof of the noncompact case and get:

Corollary 4.3. Let (Σ, σ) be a compact surface. Denote δ, h and K its systole, Cheeger constant and supremum of curvature. There is a constant C depending only on δ, h, K such that, for any u ∈ W 1,2 (Σ), satisfying:

Σ |∇u| 2 ≤ 1 (4.12) σ(u ≥ 0) = σ(u ≤ 0) (4.13) we have (4.14) Σ exp 4πu 2 -1 ≤ C
Proof. Consider u, and its radial rearrangements u + ,u -defined by theorem 4.2. As the disk is conformal to a disk of curvature -h with conformal bound depending only on r 0 , h, K, one can assume D is a disk with constant curvature -h. We can then extend by zero the maps u +m and mu -to get functions in W 1,2 on the space of constant curvature -h. Then , by [START_REF] Mancini | Moser-trudinger inequality on conformal discs[END_REF], we have C depending on δ, h, K such that:

D exp 4π(u + -m) 2 -1 ≤ C (4.15) D exp 4π(u --m) 2 -1 ≤ C (4.16)
The properties of the radial rearrangements are then enough to get the desired result: (4.17)

Σ exp 4π(u -m) 2 -1 ≤ C
In order to get the same statement for zero average functions, we will need two technical lemmas.

Lemma 4.4. Let (Σ, σ) be a finite volume surface. Denote Λ its spectral gap. Let u ∈ W 1,2 (Σ). Denote u its average and m its median. Then

(4.18) |m -u| ≤ 2Λ Vol(Σ) |∇u| 2
Proof. Recall the Markov inequality, for ε > 0: .

(4.19) σ({|u -u| ≥ ε}) ≤ ε -2 |u -u| 2 2 ≤ ε -
√ 1-m 2 A (u + -m) to get (4.34) D exp 4π 1 -4Λm 2 A (u + -m) 2 -1 ≤ C(δ, h, K) We compute (u -m) 2 1 -4Λm 2 A -u 2 = 4Λm 2 A 1 -4Λm 2 A u 2 - 2u 4ΛmA + 1 4ΛA (4.35) ≥ - 1
Hence we can bound

D exp 4πu 2 --1 ≤ C(δ, h, K)e m 2 + Vol(Σ) 2 (e m 2 -1) (4.42) ≤ C(δ, h, K)e 2Λ Vol(Σ) + Vol(Σ) 2 (e 2Λ Vol(Σ) -1) (4.43)
All in all, we get nonnegative constants C ′′ , a, b depending on δ, h, K so that:

(4.44) D exp 4πu 2 -1 ≤ C ′′ (exp a Vol(Σ) ) + Vol(Σ)(exp b Vol(Σ) -1)
The map t → t(exp b t -1) is nonincreasing on R + -{0}, hence we get an upper bound which is nonincreasing in Vol(Σ), as asserted.

We can see that these dependencies are actually sharp for the case of hyperbolic closed surfaces: Proposition 4.6. Let ε n → 0 and let (X n ) be a family of hyperbolic closed surfaces with systole ε n . Then there is a family of functions f n ∈ W 1,2 (X n ) satisfying: Proof. Fix R 0 > 0. Thanks to the collar lemma (see [START_REF] Buser | Geometry and spectra of compact Riemann surfaces[END_REF], theorem 4.1.1), if the systole goes to zero, for ε small enough X n contains an isometric collar (-R 0 , R 0 ) × S 1 with metric dt 2 + ε 2 n ch(t) 2 dθ 2 . Consider the map: (4.47)

g n (t, θ) = 1 (-R 0 ,R 0 ) (cosh(t) -cosh(R 0 )

It belongs to W 1,2 (X n ), and satisfies: In particular, if we consider a family with lower bounded volume and systole, and uniformly bounded curvature and Cheeger constant, we get a Moser-Trudinger bound independent of the surface considered in the family.

Corollary 4.7. Let X be a closed hyperbolic surface, and ε > 0 Consider X the collection of Riemannian covers of X with Cheeger constant bigger than ε. Then there is a constant C(ε) such that, for any Y ∈ X , for any u ∈ W 1,2 (Y ) satisfying: Remark 4.2. Now depending on ε this family has different characteristics: For ε > h 1 (X), this family is empty. The result of [START_REF] Magee | A random cover of a compact hyperbolic surface has relative spectral gap 3 16-ε[END_REF] combined with Buser's control tells us that for ε close enough to 0, the family is infinite, and even "big", in the sense that the probability that a random cover of degree d doesn't belong to X goes to zero as d is large.

It is not known to the author if a random tower of covers of X has a nonzero probability to belong to some X (ε), for ε > 0.

  Figure 2.1. graph of g for (K, h, ε) = (-1, 1.6, 0.2)

  Theorem 3.4. Let Σ be a (noncompact) surface with Cheeger constant 0, and lower bounded curvature. Then the Moser-Trudinger inequality is false. Proof. Thanks to the work of Buser [Bus82], we know that such a surface has no spectral gap, hence it cannot satisfy a Moser-Trudinger inequality. Remark 3.1. There are examples of surfaces from Buser [Bus79] of surfaces with Cheeger constant arbitrarily small with regard to the spectral gap, but these examples involve small systoles and very negative curvature. It is not known to the author if with those examples one could show a noncompact surface satisfying the Moser-Trudinger inequality, but with Cheeger constant zero. Theorem 3.5. Let Σ, σ be a (noncompact) surface with curvature κ satisfying: (3.23) sup κ = ∞ Then the Moser-Trudinger inequality is false.

  27) So the Moser-Trudinger inequality is false. When the systole goes to zero, there are plenty known examples where the inequality is false. Theorem 3.6. Let Σ be the hyperbolic cusp: R × S 1 equipped with the metric dt 2 + e 2t dθ 2 . Then it does not satisfy the Moser-Trudinger inequality. Proof. Fix b ∈ R, and consider the function (3.28) f (t, θ) = (e te b )1 t≤b One can make the following computations: Moser-Trudinger inequality.

  Xn|f n | 4 → +∞

  Xn|g n | 2 = 2ε n ( sinh(R 0 3 3 + sinh(R 0 ) -R 0 cosh(R 0 )) (4.50)A thorough computation gives, when ε n → 0 and R 0 too, with ε n negligible with regard to R 0 , fix R 0 = ε 1 10 small enough and you get, as n → +∞: (4.52)|g ng n |

  2 -1) ≤ C(ε)

  Lemma 4.5. Let D be a disk of area A, and spectral gap Λ. Let m > 0. Let v be a radial nondecreasing map satisfying: Proof of theorem 4.1. In order to get the result for zero average functions, we to be more careful. Let Σ be a compact surface, and consider u a Morse function on it satisfying:

	as asserted.
	(4.23)	v(∂D) = m
	(4.24)	D	v ≤ -mA .
	Then
	(4.25)	|∇v| 2 2 ≥ 4Λm 2 A
	Proof. First, we use Cauchy-Schwarz for the estimate:
	(4.26) We deduce that	2 A 2 m 2 ≤ A|v| 2
	(4.27) Hence, by definition of the spectral gap: 4A 2 m 2 ≤ |v -m| 2 2 .
	(4.28) as asserted.	|∇v| 2 2 ≥ 4ΛA 2 m 2
	(4.29)			u = 0
				Σ
	(4.30)	Σ	|∇u| 2 ≤ 1
	Up to replacing u by -u, we can assume its median satisfies: (4.31) m > 0
	Consider its radial rearrangements u (4.32) where A denotes the area of our disk. Of course, this implies |∇u -| 2 2 ≥ 4Λm 2 A
	We apply it to (4.33) We can now apply the Moser-Trudinger inequality to |∇u + | 2 2 ≤ 1 -4Λm 2 A	1	2 Λ|∇u| 2 2
	(4.20)	ε 2 =	2Λ|∇u| 2 2 Vol(Σ)
	in order to get
	(4.21)	σ({|u -u| ≥ ε}) ≤	Vol(Σ) 2	.
	By definition of the median, necessarily
	(4.22)	|u -m| ≤ ε

-,u + on the disk D, defined by theorem 4.2. As the disk is uniformly conformal to a hyperbolic disk, it has a spectral gap Λ(δ, h, K) As u is of zero average, u -satisfies the assumptions of lemma 4.5, and we have:


Thierry Bouche
email: thierry.bouche@univ-grenoble-alpes.fr

Olivier Labbe
email: olivier.labbe@univ-grenoble-alpes.fr

The New Numdam platform

The Numdam French digital mathematics library has now been in operation for more than 15 years with no major upgrade. It holds more than 57000 documents either scanned or born digital (spanning over one million pages). The information system has been recently completely redesigned. In this article, we present the new Numdam ecosystem. A metadata factory is used to store metadata from a variety of sources, normalize it under JATS (articles) or BITS (books) XML formats, and enhance it through manual editing or automated agents (tagging math formulas, matching to external databases and interlinking, etc.). The data model supports the main types of documents currently expected to populate the DML: journals, seminars, conference proceedings, multivolume works, books, book parts, doctoral theses. All documents are in collections that can belong to one or more corpus. The workflow has been simplified and allows easy deployment on test and production web sites. A platform holds all the data in one place, can generate multiple web sites, each with a different view on the data, and provides an OAI-PMH server to the outside world. Finally, the article presents future plans to create a DML-ready platform based on the new Numdam platform.

Introduction

The Numdam programme [START_REF] Bouche | Toward a digital mathematics library?[END_REF] was launched at the very beginning of this century. It started at Cellule Mathdoc as a pilot digitization project of 5 serials with the goal to provide digital preservation and wide access to our mathematical heritage. The paper collections were borrowed from publishers or libraries, the digitization itself was outsourced. At that time, it was also considered to outsource the online posting as Mathdoc felt it had little resources to manage it. However, while the collections were under development, Claude Goutorbe tested the possibility to build a web site on top of Numdam metadata based on the EDBM technology that he had developed for the first web site of the Zentralblatt MATH database. EDBM (supposedly an acronym for European database manager) is basically an indexing engine, with a library for searching, extended progressively with services written in Python. As it appeared very rapidly that this was going to work pretty well, and enabled us to keep access free to the Numdam content by avoiding external costs, the Numdam web site was open to the public back in 2002, based on an evolution of EDBM.

Numdam has evolved to become the French digital mathematics library, with more than 1 million pages acquired from a number of sources (our own digitization amounts to about 70 % of the content, partnering publishers provide born digital files and metadata for the recent articles). Last year, the web site and associated tools such as the OAI-PMH server were still based on the 15 years old EDBM. Also, a custom internal XML format was used to store metadata, meant to capture the metadata we collected for the first digitized series, with some extensions added along the way, but still designed exclusively for articles published in journals. This became more and more troublesome as we were adding different content types such as books or theses.

For instance, when Numdam participated in the EuDML project [START_REF] Bouche | Reviving the Free Public Scientific Library in the Digital Age? The EuDML project[END_REF], we had to convert our metadata to the EuDML metadata schema [START_REF] Bouche | The EuDML metadata schema: Version 1.0[END_REF] based on the NLM Journal Archiving and Interchange Tag Suite (JATS), and set-up a separate OAI-PMH server [START_REF]Open Archives Initiative: Protocol for Metadata Harvesting[END_REF] in order to expose Numdam content to the EuDML harvester.

Work on a full redesign of the Numdam hosting and dissemination platform has started a couple of years ago, initially led by the late Claude Goutorbe, now by the second author. An entirely new workflow has been built, with components dedicated to specific tasks and communication protocols between them. These components have now been released and are used in production. The Numdam web site [START_REF] Numdam | The French Digital Mathematical Library[END_REF] is running the new software since February 2017.

This article presents the new Numdam ecosystem: its architecture, the concept of a platform and its advantages to enable the construction of a larger virtual library, and the software engineering practices applied to this project to improve quality and prepare future changes.

Numdam architecture

The overall Numdam architecture is presented in Figure 1.

The Metadata Factory

A separate application, the metadata factory, shown on Figure 2, imports and enhances metadata in the overall system. Documents from a variety of sources (digitized by Mathdoc, by other partners, or born digital documents) are all ingested in the metadata factory. A web interface allows manual modification of any metadata snippet (see figure 3). A number of services that enhance metadata can also be launched selectively on a range of items. The metadata factory software is based on eXistdb, which is a native XML database system [START_REF]Exist Solutions: eXistdb, an Open Source native XML database[END_REF], with editing templates adapted to our use of JATS and BITS.

This is where we perform the Extract Transform Load (ETL) steps (see, e.g. [22]). In our case, this means that for each ingested item, the XML is either loaded without change if it is tagged according to the new internal format (based on JATS for journal articles [START_REF]National Center for Biotechnology Information: Journal archiving and interchange tag library[END_REF], and BITS for books [START_REF]Book interchange tag suite (BITS)[END_REF]), or transformed on the fly to that format if it is tagged under our legacy custom DTD. For metadata coming from publishers, we typically perform transformations to JATS before ingestion.

An anti-pattern is a common structure or set of actions that initially looks like an appropriate and effective solution, but leads you eventually into trouble [START_REF] Koenig | Patterns and antipatterns[END_REF]. As noted by Ken Farmer in his list of ETL anti-patterns [START_REF] Farmer | Data Warehouse ETL for Data Scientists[END_REF], a single program must not be used to perform all the steps. Therefore, we developed and use separate automated agents to enhance metadata:

we use a spell checker in order to find OCR or typing errors in metadata such as titles or author names. we have a routine to correct some usual orthotypographical mistakes such as abusive capitalization or wrong spacing. we have separate routines to look up external databases such as EuDML, Math-SciNet or zbMATH in order to provide deep interlinking for our articles or citations, we generate L A T E X driver files that produce the PDF full texts with cover pages.

as we input all mathematical formulas as L A T E X, they are converted on the fly to MathML and stored in the XML file next to the T E X code, we also have a number of programs looking up our metadata for possible problems or errors (such as a relation from errata without backward relation. . . ). In this way, all the ingestion, manual or automated enhancements are managed in the metadata factory, which holds our reference metadata, which is fed to the public web site through a REST API.

The Numdam web site

The Numdam web site is based on a standard architecture:

metadata are stored in a relational database, the Solr search engine is used to search keywords and provide facets in the search results, a file server returns the documents to the user (PDF, DjVu). A protection mechanism prevents users from downloading the full text of the most recent articles for journals following a moving wall principle, a web application, developed using the popular framework Django, displays the Numdam web pages in a browser interface, an OAI-PMH server exports metadata in multiple formats: the mandatory simple oai-dc, eudlm-article2 and eudml-book2 EuDML formats for much more details, and gallica (which exposes journal-level metadata in oai-dc).

The legacy Numdam (see Figure 4) was based on a custom made search engine in the past. All metadata was stored in a PostgreSQL database with a structure modelled on our legacy (journal-only) DTD. All dynamical pages were produced on the fly as a result of a search query: The search and browsing interfaces, as well as the items' short and long record display pages were generated as a response to a search query. The data returned by the database was assembled (and transformed with XSLT) in order to build the page content. We didn't have a metadata factory but metadata was prepared using various scripts and manual editing yielding a collection of XML files that were ingested and indexed by EDBM. EDBM also created special indexes for features that were not implemented in the XML files (such as reverse citations, links to citing items from a cited item, etc.). We also had a special trouble coming from the fact that the content managing system SPIP [START_REF]SPIP, a publishing system for the Internet[END_REF] was used for the static web pages at www.numdam.org. The dynamic part of the web site had to mimick SPIP pages and menus, with hard coded links to SPIP articles. As SPIP and EDBM didn't manage some features the same way (like, for instance, bilingualism: cookies in the cas of SPIP, parameters in the case of EDBM), this could lead to a quite uneven user experience such as launching a search query in the English interface and ending up in a page written in French. . . The new search engine is based on the Lucene search library. Communication with the Lucene engine is based on standards like XML, JSON and HTTP. Solr follows the NoSQL (Not only SQL) movement : the data stores are non-relational and therefore do not require fixed table schemas. They avoid join operations to retrieve documents, improving the search operation for complex queries. Solr brings additional benefits, such as faceting, spell checking, similar item search, hit highlighting and free text search. XSLT (EXtensible Stylesheet Language) was used to transform the model into XHTML, which was fine when user interactions were few and pages did not react to the environment, like the size of the browser window. The Model-View-Controller (MVC) design pattern [START_REF] Gamma | Design Patterns: Elements of Reusable Object-oriented Software[END_REF] is now applied to separate the view from the user interactions, which are more important today with search facets used to filter search results. Graphic designers can now focus on the ergonomics of the web site, and can provide an adaptive view that reacts to the size of the window.

For a user, the Numdam interface looks like a common one for a digital library: menus allow navigation across collections, search fields let you specify search keywords, and facets let you quickly narrow the search results (see Figure 4). In the past, Numdam objects were built to model a journal. Hacks had to be put in place to support memoirs and doctoral theses. The fact that we now have native support for other document types than journal articles has been used to have a different item page layout (aka landing page) depending on the item type (books or theses, edited books, chapters in edited books. . . ). We are also experimenting with a specific browsing interface for books, ordered as in library shelves by author, year, title. All the content is now organized in three major document types: journal article, single book, and multiple volume book. In Numdam, we have composed our objects into tree structures to represent a part-whole hierarchy, thus following the Composite design pattern in Object-Oriented software [START_REF] Gamma | Design Patterns: Elements of Reusable Object-oriented Software[END_REF]. Figure 5 shows the objects used in our model: leaf: journal article or book part, container: journal issue, book (including monograph and edited book), collection: journal, book series, multivolume work. As a collection is a composite, it can be embedded in a larger collection. For example, the Séminaire Bourbaki is a collection available in Numdam that happens to have been published in different venues over time (including Springer Lecture Notes in Mathematics, SMF's Astérisque series). A subset of this collection is thus part of the Astérisque collection, whose other volumes are currently being digitized and will soon be added to Numdam.

Mathematical formulas

The handling of mathematical formulas has been improved thanks to the new system. In the previous Numdam system, we encoded metadata using an XML DTD that didn't know anything about mathematical content. As a result, we stored T E X math encoding (with dollars) as text strings (with special XML characters escaped). Thus the math in Numdam, although a reference math library, was not handled as such, and not very much exploitable as math knowledge. We did tag formulas as math and convert most of them to MathML when we generated JATS XML to be ingested in EuDML, but we could not exploit this on our own site! Now the metadata factory automatically detects L A T E X math environments while editing text blocks that may contain it (such as title, abstract, keyword. . . ). The JATS XML superstructure identifying formulas is added, and the MathML alternative to the T E X code is added. Team members without computational skills can now use the metadata factory, edit mathematical formulas and immediately preview the graphical form. Automated tests have been written to verify the validity of the T E X source code in all the articles stored in Numdam. We were able to detect and fix a little less than 100 issues. The improvement in the mathematical formulas handling allowed us to set additional goals and support mathematical formulas in references.

By default, mathematical formulas are displayed in graphical form using Math-Jax from MathML (see Figure 6). A switch, easily accessible in all page headers, lets you display the T E X source code of the formulas for accessibility or copy-pasting (see Figure 7). These changes allow us to be fully embedded in the network of DMLs as we are now able to serve MathML (and, in fact, some other math-specific goodies such as mathematical subject classification codes) and full EuDML compliant metadata through our OAI-PMH server.

Numdam platform

The new Numdam has been designed so that the same platform can generate multiple sites, all of them using the same data (PDF, DjVu, images, metadata, search engine). The industry platform definition is used here: "a set of assets organized in a common structure from which a company can efficiently develop and produce a stream of derivative products" [START_REF] Gawer | Industry platforms and ecosystem innovation[END_REF].

The platform development was initially started in order to provide Mathdoc with a versatile tool for storing and delivering documents that we host like the Numdam collections, as well as some companion collections that will be added to the Numdam web site as "associated libraries".

An important feature of the platform is that it is very easy to update the platform content through its REST API, and the new system supports incremental updates (at the granularity of journal issue or single book), which will simplify a lot the online posting workflow as the (non-technical) people using the metadata factory can check their editing on a test web site until they are happy with it, then switch the validated items to the public web site. This is in strong contrast with the previous EDBM-based system where a large global index had to be recomputed from scratch for each modification.

We are also currently working on an extension to the platform that will replace the current farm of EDBM-powered Cedram web sites (where we have one overall web site with a global search engine, and one web site for each journal with a specific layout and web design, see [START_REF] Bouche | CEDRICS: When CEDRAM meets Tralics[END_REF]). As each journal publishes regularly content indepedant from each other, the possibility to add or remove articles while preparing new issues will be of great value to our staff.

Figure 8 shows 3 sites: numdam.org and 2 journal's web sites (aif.cedram. org and afst.cedram.org) that can be based on the same platform. The 3 sites can be hosted on the same physical server, or they can be cloned on separate virtual machines if needed, to improve performance.

An advantage of such an architecture is that bibliographical data can be collated and curated in only one place. Harvesting the metadata gathered by Mathdoc in a bigger virtual library is facilitated. As we have much better quality control on our metadata and native support to serve full math-aware metadata, the Numdam platform helps furthering the DML objective, this time using the DML definition of the "virtual library of all mathematical texts available digitally, both retrodigitized and born digital" [START_REF] Bouche | Introducing the mini-DML project[END_REF]. As all document types envisioned to form the DML are natively supported by the platform, it can also serve to run a large DML system for content delivery, with content harvested through OAI-PMH or other means, this is a goal we intend to pursue in a near future.

In order to make the new Numdam platform successful, we had to prepare the change. The first step was to realize that Numdam was not just a software application, but an information system: a "Formal, sociotechnical, organizational system designed Fig. 8. The New Numdam platform: multisite. to collect, process, store, and distribute information" [START_REF] Piccoli | Information Systems for Managers: Texts and Cases[END_REF]. An information system has four components: technology, structure, people and process. Designing an information system requires to consider each component, not just new features or new technologies. In the case of Numdam, a secondorder change was anticipated. In a second-order change, the technology and processes evolve, and the people who perform the processes are affected by the change [START_REF] Piccoli | Information Systems for Managers: Texts and Cases[END_REF] (see Figure 9). The objective was not only to replace the technology to automate some tasks or to use a more modular and modern technology, but also to involve a different set of people. The change was anticipated, and attention was paid to the ease of use of the interface. The metadata factory dashboard displays tasks that can be launched from the web interface, the status of these tasks, and lets you edit any metadata. The interface is now meant to be used by nontechnical people. As a result of this second-order change, metadata are now updated more efficiently. They can be detected quickly and corrected before publication, by team members without computational skills.

To ensure that the interface of both the metadata factory and the Numdam web site were intuitive, an agile methodology, based on the SCRUM development process [START_REF] Schwaber | Scrum development process[END_REF] was followed. Features were delivered on a regular basis and presented to the users. This iterative approach allowed us to implement the most important features first and to collaborate with the users regularly, thus following one of the principles of the Agile manifesto [START_REF] Beck | Manifesto for agile software development[END_REF] where "customer" collaboration is valued over contract negotiation. Rewriting Numdam was also an opportunity to introduce some of the software engineering best practices. Test-driven development (TDD), where test cases are created before the code is written, helps prevent defects much more efficiently than many other forms of testing [START_REF] Jones | Software Engineering Best Practices[END_REF]. Continuous delivery, in particular deployment automation, reduces the delivery cycle time, and hence gets bug fixes to users faster [START_REF] Humble | Continuous Delivery: Reliable Software Releases Through Build, Test, and Deployment Automation[END_REF]. This practice enabled us to deploy the source code and all the data (binary files, metadata) to test or production servers easily. In addition, it fostered the development team to submit and deploy the changes frequently, thus iterative development became natural. These software engineering best practices not only improve the software quality, but will also enable us to deliver multiple sites (Numdam, Cedram). The architecture will also make it easier to develop more APIs as the need for them appears, e.g. in the context of an international coordination of a global DML such as envisioned by the International Mathematical Knowledge Trust (IMKT, see [START_REF] Ion | The Global Digital Mathematics Library and the International Mathematical Knowledge Trust[END_REF]).

Conclusion

The new Numdam platform, publicly available since early 2017, was designed with a standard architecture (database, search engine, file server, OAI-PMH server, web applications developed with a popular framework). Some of the software engineering best practices were applied which make the source easier to maintain, and the model easier to extend. This platform will enable us to efficiently develop and produce a stream of derivative web sites. The next objective is to support the Cedram family of journals and to add associated collections to Numdam: Gallica-math [START_REF]Gallica-Math, a front-end to some mathematical content from Gallica[END_REF] and Orsay mathematical publications [START_REF][END_REF] for a start. These collections are currently being posted with ad hoc unmaintained applications, and will benefit of much more visibility once integrated into the Numdam platform. Moreover they will be easily contributed to any DML harvester, thus enlarging the rich content already available in, e.g. EuDML.

In this sense, we view the platform as an enabling device to contribute more content to the global DML: We will be able to host in the same system, with the same quality metadata and a versatile format supporting all document types, more collections, including those from institutions that do not have the capacity to properly make them visible and interoperable to the other DML projects (or, in the case of very large generic digital libraries such as Gallica, the Digital Library of the French National Library, that would not allocate resources to a math-only project such as curating mathematical metadata), Another application of the platform will be to create a DML web site much larger than those already existing. The goal is to create a larger virtual library, by harvesting metadata from other digital libraries, with much more detailed metadata than was available in the mini-DML [START_REF] Bouche | Introducing the mini-DML project[END_REF], and a much larger coverage. This will require to set up an OAI-PMH harvester using the REST API in order to populate that instance of the platform.

Fig. 1 .

 1 Fig. 1. Numdam platform architecture.

Fig. 2 .

 2 Fig. 2. The metadata factory dashboard.

Fig. 3 .

 3 Fig. 3. The metadata factory: editing XML metadata.

Fig. 4 .

 4 Fig. 4. The old Numdam web site.

Fig. 5 .

 5 Fig. 5. Part-whole hierarchy.

Fig. 6 .

 6 Fig. 6. Mathematical formulas in graphical form.

Fig. 7 .

 7 Fig.7. T E X source code of the formulas.

Fig. 9 .

 9 Fig. 9. Second-order change.

This article is available at Zenodo via https://dx.doi.org/10.5281/zenodo. 581405.

We do believe that the architecture of the platform will make it easy to build such a virtual library, and to evolve with new APIs and innovative services on top of its holdings in order to keep pace with advances in the DML.