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

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 $\lambda_1(ds^2) Area(ds^2) \leq 16(4 - \sqrt{7})\hspace{.02cm}\pi \approx 21.668\hspace{.02cm}\pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\hspace{.02cm}\pi$.