First eigenvalue of the Laplacian on compact surfaces for large genera

TL;DR

This paper improves the upper bound on the asymptotic growth of the normalized first eigenvalue of the Laplacian on compact surfaces as genus increases, showing it is at most approximately 3.056 times pi.

Contribution

It refines the known asymptotic bound for the supremum of the first Laplacian eigenvalue on high-genus surfaces, providing a tighter limit than previous results.

Findings

The limsup of the normalized eigenvalue growth is at most approximately 3.056 pi.

The result improves upon the previous bound of 4 pi.

As genus increases, the eigenvalue growth rate is tightly bounded.

Abstract

For any Riemannian metric $ds^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian $\lambda_1(ds^2)Area(ds^2)$ 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 \[ \limsup_{g\, \rightarrow\, \infty} \, \frac{1}{g}\Lambda_1(g) \leq 4(3-\sqrt{5})\pi \approx 3.056\pi. \]