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How large can the first eigenvalue be on a surface of genus two?

D. Jakobson·M. Levitin·N. Nadirashvili·N. Nigam·I. Polterovich
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Abstract

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface which is realized as a double branched covering over a sphere. The six ramification points are chosen in such a way that this surface has a complex structure of the Bolza surface. We prove that our conjecture follows from a lower bound on the first eigenvalue of a certain mixed Dirichlet-Neumann boundary value problem on a half-disk. The latter can be studied numerically, and we present conclusive evidence supporting the conjecture.

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D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam, I. Polterovich. 2005-09-18. How large can the first eigenvalue be on a surface of genus two?. https://arxiv.org/abs/math/0509398

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