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

• Published in: Geometric and functional analysis Vol. 24; no. 4; pp. 1336 - 1376
• Main Author: Petrides, Romain
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.08.2014
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Klein Bottle; Conical Singularity; Hyperbolic Surface; Extremal Function; Minimal Surface
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-014-0292-5

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 (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013 ). 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 (Int Math Res Not 17:939–952, 1999 ) 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.