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...

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