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

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 $\approx 21.414\,\pi$.