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

Abstract

For any compact riemannian surface of genus three $(\Sigma ,d{s}^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian ${\lambda }_1(d{s}^2)$ and the area $Area(d{s}^2)$ is bounded above by $24\pi$. In this paper we improve the result and we show that ${\lambda }_1(d{s}^2)Area(d{s}^2)\le 16(4-\sqrt{7})\pi \approx 21.668\pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\pi$.

arXiv commentFinal version. To appear in J. Math. Soc. Japan