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.

中文翻译：

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关于第一个Laplace特征值的Yang-Yau不等式

在1980年发表的一篇开创性论文中，P。C. Yang和S.-T. 丘证明了一个不等式，根据其属（\ gamma \）和面积，限制了拉普拉斯算子在可定向黎曼曲面上的第一个特征值。J. Hersch的旧结果对\（\ gamma = 0 \）实现了Yang–Yau估计的等式，并且S. Nayatani和T. Shoda最近证明，\（\ gamma = 2 \）。在本文中，我们将代数几何和最小曲面理论的技术结合起来，证明Yang-Yau的不等式对于所有（\ gamma> 2 \）都是严格的。以前仅以\（\ gamma = 1 \）闻名。在本文的第二部分中，我们应用Chern-Wolfson的谐波序列概念来获得谐波映射从表面到球体的总分支阶数的上限。讨论了这些结果在特征值极值度量中的应用。