Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

Abstract. We investigate in this paper the existence of a metric which maxi-mizes 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 isomet-ric 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. Let (Σ, g) be a smooth compact Riemannian surface without boundary. The eigenvalues of the Laplacian ∆g = −divg (∇) form a discrete sequence 0 = λ0 < λ1 (Σ, g) ≤ λ2 (Σ, g) ≤... Getting bounds on these eigenvalues depending on the metric or the topology of Σ has been the subject of intensive studies in the past decades. In this paper, we shall focus on the first eigenvalue λ1. One can for instance consider the first conformal eigenvalue of (Σ, g) defined by Λ1 (Σ, [g]) = sup g̃∈[g] λ1 (g̃)V olg ̃ (Σ). (0.1) If one looks at the infimum of the first eigenvalue in a given conformal class, it is always 0. Now one can also study invariants which depend only on the topology of the surface. For orientable surfaces, one can define for any genus γ ≥ 0 Λ1 (γ) = sup g λ1(g)V olg (Σ) = su