论文标题
laplacian的第一个特征值大属地面紧凑型表面
First eigenvalue of the Laplacian on compact surfaces for large genera
论文作者
论文摘要
对于任何Riemannian Metric $ ds^2 $在紧凑型属$ g $的表面上，杨和杨证明，laplacian $λ_1（ds^2）区域（ds^2）$的归一化第一个特征值在该属中是有限的。特别是，如果$λ_1（g）$是每个$ g $的至高无上的，则得出的是，序列$ {λ_1（g）} $的渐近生长不超过$4πg$之一。在本文中，我们改善了结果，并证明\ [\ limsup_ {g \，\ rightArrow \，\ infty} \，\ frac {1} {g} {g}λ_1（g）\ leq 4（3- \ sqrt {5}）π\近3.056π。 \]
For any Riemannian metric $ds^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian $λ_1(ds^2)Area(ds^2)$ is bounded in terms of the genus. In particular, if $Λ_1(g)$ is the supremum for each $g$, it follows that the asymptotic growth of the sequence ${Λ_1(g)}$ is no larger than the one of $4πg$. In this paper we improve the result and we show that \[ \limsup_{g\, \rightarrow\, \infty} \, \frac{1}{g}Λ_1(g) \leq 4(3-\sqrt{5})π\approx 3.056π. \]
