Second Variation Formula for Eigenvalue Functionals on Surfaces

TL;DR

This paper derives the second variation formula for eigenvalue functionals on surfaces, providing criteria to identify local maximizers and applying it to show the flat metric on certain tori cannot maximize the first eigenvalue.

Contribution

It introduces the second variation formula for eigenvalue functionals on surfaces, extending previous first-order results to analyze local maximizers.

Findings

Flat metric on non-rhombic torus is not a conformal maximizer for the first eigenvalue.

Second variation formula helps determine local optimality of eigenvalue metrics.

Results extend to Steklov eigenvalues on cylinders.

Abstract

Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional and minimal surfaces in the sphere. It was later extended by A. El Soufi and S. Ilias to cover k-th eigenvalues and critical points in a fixed conformal class, where the latter correspond to harmonic maps to the sphere. These results, however, only contain first order information and cannot be used to determine whether a given critical metric a local maximiser or not. In the present paper we write down the second variation formula for critical metrics and show that the flat metric on the non-rhombic torus can never be a conformal maximiser for the first eigenvalue. Analogous results are proved in the context of the Steklov eigenvalues and flat metrics on a cylinder.