On the first eigenvalue of the laplacian on compact surfaces of genus three

TL;DR

This paper improves an upper bound on the product of the first Laplacian eigenvalue and area for genus three surfaces, providing a tighter estimate and numerical evidence near the bound.

Contribution

The authors refine the upper bound for the first eigenvalue-area product on genus three surfaces, surpassing previous results by Yang and Yau.

Findings

New upper bound: approximately 21.668π.

Numerical computations for the Klein quartic surface yield about 21.414π.

The bound is close to the numerical value for the hyperbolic Klein quartic.

Abstract

For any compact riemannian surface of genus three $(\Sigma,ds^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian $\lambda_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24\pi$. In this paper we improve the result and we show that $\lambda_1(ds^2)Area(ds^2)\leq16(4-\sqrt{7})\pi \approx 21.668\,\pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\,\pi$.