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In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved aninequality bounding the first eigenvalue of the Laplacian on an orientableRiemannian surface in terms of its genus $\gamma$ and the area. The equality inYang-Yau's estimate is attained for $\gamma=0$ by an old result of J. Herschand it was recently shown by S. Nayatani and T. Shoda that it is also attainedfor $\gamma=2$. In the present article we combine techniques from algebraicgeometry and minimal surface theory to show that Yang-Yau's inequality isstrict for all genera $\gamma> 2$. Previously this was only known for$\gamma=1$. In the second part of the paper we apply Chern-Wolfson's notion ofharmonic sequence to obtain an upper bound on the total branching order ofharmonic maps from surfaces to spheres. Applications of these results toextremal metrics for eigenvalues are discussed.

On the Yang-Yau inequality for the first Laplace eigenvalue