Second Variation Formula for Eigenvalue Functionals on Surfaces

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.