Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces∗
1 Department of Mathematical Sciences, Claremont McKenna College, CA 91711, USA.
2 Department of Mathematics, Rensselaer Polytechnic Institute, NY 12180, USA.
3 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
Received: 8 March 2014
Revised: 5 September 2015
Accepted: 29 January 2016
Let (M,g) be a connected, closed, orientable Riemannian surface and denote by λk(M,g) the kth eigenvalue of the Laplace−Beltrami operator on (M,g). In this paper, we consider the mapping (M,g) → 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) 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 Λtk(γ) . Our computations support a conjecture of [N. Nadirashvili, J. Differ. Geom. 61 (2002) 335–340.] that Λtk(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 Λtk(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 λk = 4π2 ⌈k/2⌉2(⌈k/2⌉2 − 1/4)-1/2. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.
Mathematics Subject Classification: 35P15 / 49Q10 / 65N25 / 58J50 / 58C40
Key words: Extremal Laplace−Beltrami eigenvalues / conformal spectrum / topological spectrum / closed Riemannian surface / spectral geometry / isoperimetric inequality
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