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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94641Sharp 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.20 pages; 4 figuresSpectral TheoryMetric Geometry35P15; 47A75How large can the first eigenvalue be on a surface of genus two?text