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We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen [16]. Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili [17] which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface. √ 3 and the maximal metric is given by the flat equilateral torus (see [36]). At last, the genus 2 case was recently obtained : we have Λ 1 (2) = 16π and there is a family of maximal metrics (see [26]).The spectral gap Λ 1 (γ) > Λ 1 (γ − 1) necessarily holds for an infinite number of γ thanks to the lower bound (0.4). It is believed to hold for all genuses. The extremal metric in the theorem is the pull-back of the induced metric of a minimal immersion (with branched points) of Σ into some sphere S k . As a classical corollary of the above theorem, we obtain the following :, which is the case at least for an infinite number of γ, there exists a minimal immersion (possibly with branch points) of a compact surface Σ of genus γ into some sphere S k by first eigenfunctions.There have been lot of works about minimal immersions of surfaces into spheres. In particular, they are necessarily given by eigenfunctions (not only first eigenfunctions) thanks to Takahashi [41]. For existence results of such immersions, we refer to two classical papers by Lawson [29] and Bryant [4]. Concerning minimal embeddings in S 3 , it is conjectured by Yau [44] that they all come from first eigenfunctions (see [2] and [8] for recent surveys on this subject). However, minimal immersions by first eigenfunctions are not so numerous. For instance, it has been proved by Montiel and Ros [33] that there is at most one minimal immersion by first eigenfunctions in any given conformal class. In the case of genus 1, it was also proved by El Soufi and Ilias [14] that the only minimal immersions by first eigenfunctions of the torus are the Clifford torus (in S 3 ) and the flat equilateral torus (in S 5 ). So our corollary is interesting because it provides an infinite number of new minimal immersions into spheres by first eigenfunctions.At last, we prove a conjecture stated in [17] about the infimum of the first conformal eigenvalue on any orientable surface :Theorem 3. Let Σ be a smooth compact orientable surface. Then inf [g] Λ 1 (Σ, [g]) = 8π and this infimum is never attained except on the sphere. This result had already been proved in [18] in genus 1...

We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of R n+1 with bounded energy. This generalizes results obtained by Da Lio [1] on the disc.Let (M, g) be a smooth Riemannian surface with a smooth nonempty boundary with s connected components. We fix n ≥ 2 and let B n+1 be the unit ball of R n+1 . A map u : (M, g) → B n+1 is a smooth harmonic map with free boundary if it is harmonic and smooth up to the boundary, u(∂M ) ⊂ S n and ∂ ν u is parallel to u, (or ∂ ν u ∈ (T u S n ) ⊥ ). The energy of such a map is defined as

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lies in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Colding-Minicozzi replacement procedure.

We perform the classical Ginzburg-Landau analysis originated from the celebrated paper by Bethuel, Brezis, Hélein for optimal boundary data. More precisely, we give optimal regularity assumptions on the boundary curve of planar domains and Dirichlet boundary data on them. When the Dirichlet boundary data is the tangent vector field of the boundary curve, our framework allows us to define a natural energy minimizing frame for simply connected domains enclosing Weil-Petersson curves.1 2 (Γ) such that deg(g) = 0 and C 1 domains, but to our knowledge, the classical Ginzburg-Landau analysis coming from the originated paper [1] has never been completely given for deg(g) = 0 with g ∈ H 1 2 (Γ), nor with domains Ω with lower regularity than C 1 .

In this paper we establish an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.Given (M, g) a smooth compact Riemannian manifold (without boundary), the spectrum of the Laplacian Δ g = −div g (∇) is a discrete sequence of eigenvalueswhich goes to +∞ as k → +∞. The eigenfunctions associated to the simple eigenvalue λ 0 = 0 are the constant functions. A natural, and often addressed, question is how to get estimates on the eigenvalues thanks to some geometric assumptions. In this paper, we discuss the maximization of eigenvalues for metrics in a given conformal class with fixed volume. We focus on the case of the standard sphere.We let S n be the unit sphere of R n+1 for n ≥ 2. If g is a metric on S n , we are interested in the scale invariant quantity Λ n,k (g) = λ k (S n , g)Vol g (S n ) 2 n .In dimension 2, we can maximize Λ 2,k on regular metrics. An inequality has been proved for k = 1 by Hersch [6]: Λ 2,1 (g) ≤ 8π, with equality iff g is the round metric. He followed the proof of the maximization by Szegö [9] of the first non-zero Neumann eigenvalue for planar domains, attained by discs. Nadirashvili found an optimal maximization for k = 2. He proved inwhere the supremum is attained in the degenerate case of the union of two identical spheres. His idea was used later in [5] to show that among simply connected planar domains, the second non-zero Neumann eigenvalue is maximal in the degenerate case of two discs of the same area. If we look for an analogous inequality in dimension n ≥ 3, we have to restrict our attention to some classes of metrics since Λ n,k is not bounded on the set of regular metrics (see [2]). It is natural, as suggested in [4] and [3], to consider the set of

Abstract. Given (M, g) a smooth compact Riemannian manifold without boundary of dimension n ≥ 3, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal to g of volume 1. We prove that it is always greater than nω 2 n n , the value it takes in the conformal class of the round sphere, except if (M, g) is conformally diffeomorphic to the standard sphere.Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 3 and let us define the first conformal eigenvalue of (M, g) bywhere λ 1 (M, g) is the first nonzero eigenvalue of the Laplacian ∆ g = −div g (∇) and [g] is the conformal class of g. In this paper, we aim at proving a rigidity result concerning this first conformal eigenvalue.The maximisation on conformal classes is natural because the scale invariant quantity supremum is infinite among all metrics [3] (except in dimension 2, [16]), while El Soufi and Ilias [7] proved that it is always bounded among conformal metrics. Generalizing a result by Li and Yau [13] in dimension 2, they gave an explicit upper bound thanks to the m-conformal volumeThese conformal invariants on the standard sphere (S n , [can]) satisfy, [7]

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