First eigenvalue of the Laplacian on compact surfaces for large genera

For any Riemannian metric $d{s}^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian ${\lambda }_1(d{s}^2)Area(d{s}^2)$ is bounded in terms of the genus. In particular, if ${\mathrm{\Lambda }}_1(g)$ is the supremum for each $g$, it follows that the asymptotic growth of the sequence ${\mathrm{\Lambda }}_1(g)$ is no larger than the one of $4\pi g$. In this paper we improve the result and we show that $$\underset{g\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{1}{g}{\mathrm{\Lambda }}_1(g)\le 4(3-\sqrt{5})\pi \approx 3.056\pi \mathrm{.}$$