Heriot-Watt Mathematics Report Series
HWM05-19, 8 Sep 2005
How large can the first eigenvalue be on a surface of genus two?
D Jakobson, M Levitin, N Nadirashvili, N Nigam and I Polterovich
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 is conformally equivalent to 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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Full text: http://www.ma.hw.ac.uk/~levitin/research.html#genus
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