Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces^∗

¹ Department of Mathematical Sciences, Claremont McKenna College, CA 91711, USA.
Ckao@ClaremontMckenna.edu
² Department of Mathematics, Rensselaer Polytechnic Institute, NY 12180, USA.
lair@rpi.edu
³ Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
osting@math.utah.edu

Received: 8 March 2014
Revised: 5 September 2015
Accepted: 29 January 2016

Abstract

Let (M,g) be a connected, closed, orientable Riemannian surface and denote by λ_k(M,g) the kth eigenvalue of the Laplace−Beltrami operator on (M,g). In this paper, we consider the mapping (M,g) → λ_k(M,g). We propose a computational method for finding the conformal spectrum Λ^c_k(M,[g0]), which is defined by the eigenvalue optimization problem of maximizing λ_k(M,g) for k fixed as g varies within a conformal class [ g₀ ] of fixed volume vol(M,g) = 1. We also propose a computational 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 by Λ^t_k(γ) . Our computations support a conjecture of [N. Nadirashvili, J. Differ. Geom. 61 (2002) 335–340.] that Λ^t_k(0) = 8πk , attained by a sequence of surfaces degenerating to a union of k identical round spheres. Furthermore, based on our computations, we conjecture that Λ^t_k(1) = 8π2/√3 + 8π(k − 1) , attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and k − 1 identical round spheres. The values are compared to several surfaces where the Laplace−Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the kth Laplace−Beltrami eigenvalue has a local maximum with value λ_k = 4π² ⌈k/2⌉²(⌈k/2⌉² − 1/4)^-1/2. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.