Maximal Laplace-Beltrami eigenvalues on closed Riemannian surfaces
Chiu‐Yen Kao, Rongjie Lai, Braxton Osting · arXiv (Cornell University) · 2014
Let (M, g) be a connected, closed, orientable Riemannian surface and denote by λk(M, g) the k-th eigenvalue of the Laplace-Beltrami operator on (M, g). In this paper, we consider the mapping (M, g) 7 → λk(M, g). We propose a computational method for finding the conformal spectrum λck(M, [g0]), which is defined by the eigenvalue optimization problem of maximizing λk(M, g) as g varies within a conformal class [g0] of fixed volume vol(M, g) = 1. We also propose a method for the problem where M is additionally allowed to vary over surfaces with fixed genus, γ. This is known as the topological spectrum for genus γ and denoted λtk(γ). Our computa-tions support a conjecture of N. Nadirashvili (2002) that λtk(0) = 8pik, attained by a sequence of surfaces degenerating to a union of k identical round spheres. Furthermore, based on our computations, we conjecture that λtk(1) =