Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 685-720.

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 k c ( M , [ g 0 ] ) , which is defined by the eigenvalue optimization problem of maximizing λ k (M,g) for k fixed as g varies within a conformal class [g 0 ] 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 k t ( γ ) . Our computations support a conjecture of [N. Nadirashvili, J. Differ. Geom. 61 (2002) 335–340.] that Λ k t ( 0 ) = 8 π k , attained by a sequence of surfaces degenerating to a union of k identical round spheres. Furthermore, based on our computations, we conjecture that Λ k t ( 1 ) = 8 π 2 3 + 8 π ( 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 = 4 π 2 k 2 2 ( k 2 2 - 1 4 ) - 1 2 . Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

DOI : 10.1051/cocv/2016008
Classification : 35P15, 49Q10, 65N25, 58J50, 58C40
Mots clés : Extremal Laplace−Beltrami eigenvalues, conformal spectrum, topological spectrum, closed Riemannian surface, spectral geometry, isoperimetric inequality
Kao, Chiu-Yen 1 ; Lai, Rongjie 2 ; Osting, Braxton 3

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.
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Kao, Chiu-Yen; Lai, Rongjie; Osting, Braxton. Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 685-720. doi : 10.1051/cocv/2016008. http://www.numdam.org/articles/10.1051/cocv/2016008/

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