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

By R. Petrides. 2013

82 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d38cdc01b80b01d56c7f39efff3ac4479d776329

Citations in the corpus

• An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “∞ A number of important results on the existence and regularity of maximizing metrics has been recently obtained, see [31, 42, 43, 51].”
• SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “Similar existence results have been proved for the first nonzero eigenvalue of the Laplace–Beltrami operator in a fixed conformal class of a closed surface of arbitrary genus, in which case conical singularities have to be allowed (see [65, 89]).”