关于亏格三的紧致曲面上拉普拉斯算子的第一特征值

Pub Date : 2020-10-28 DOI:10.2969/jmsj/85898589

A. Ros

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引用次数: 10

摘要

对于任何亏格为3$（\Sigma，ds^2）$Yang和Yau的紧致黎曼曲面，证明了拉普拉斯算子$\lambda_1（ds^2）$的第一特征值与面积$area（ds ^2）$之积的上界为$24\pi$。在本文中，我们改进了结果，并证明了$\lambda_1（ds^2）Area（ds ^2）\leq16（4-\sqrt｛7｝）\pi\约21.668\，\pi$。关于界的锐度，对于双曲克莱因四次曲面的数值计算，给出了值$\约21.414\，\pi$。

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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 $\approx 21.414\,\pi$.

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