  In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an
inequality bounding the first eigenvalue of the Laplacian on an orientable
Riemannian surface in terms of its genus $\gamma$ and the area. The equality in
Yang-Yau's estimate is attained for $\gamma=0$ by an old result of J. Hersch
and it was recently shown by S. Nayatani and T. Shoda that it is also attained
for $\gamma=2$. In the present article we combine techniques from algebraic
geometry and minimal surface theory to show that Yang-Yau's inequality is
strict 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 of
harmonic sequence to obtain an upper bound on the total branching order of
harmonic maps from surfaces to spheres. Applications of these results to
extremal metrics for eigenvalues are discussed.
