First eigenvalue of the Laplacian on compact surfaces for large genera

Abstract

For any Riemannian metric $d{s}^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian ${\lambda }_1(d{s}^2)Area(d{s}^2)$ is bounded in terms of the genus. In particular, if ${\Lambda }_1(g)$ is the supremum for each $g$, it follows that the asymptotic growth of the sequence ${\Lambda }_1(g)$ is no larger than the one of $4\pi g$. In this paper we improve the result and we show that \[ \limsup_{g\, \rightarrow\, \infty} \, \frac{1}{g}Λ_1(g) \leq 4(3-\sqrt{5})π\approx 3.056π. \]