How large can the first eigenvalue be on a surface of genus two?
Spectral Theory
2007-05-23 v1 Metric Geometry
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.
Cite
@article{arxiv.math/0509398,
title = {How large can the first eigenvalue be on a surface of genus two?},
author = {D. Jakobson and M. Levitin and N. Nadirashvili and N. Nigam and I. Polterovich},
journal= {arXiv preprint arXiv:math/0509398},
year = {2007}
}
Comments
20 pages; 4 figures