紧黎曼曲面上的Kähler势谱估计
Spectral Estimates for Compact Riemann Surfaces via Kähler Potentials
- University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过Kähler势的振荡、梯度及Laplacian，定量比较紧黎曼曲面上等面积Kähler度量的谱，并在球面上改进Hersch界。
AI中文摘要：
我们建立了紧黎曼曲面上等面积Kähler度量的Laplace-Beltrami谱之间的定量比较。基于势的振荡的估计给出了倒数特征值的分数次幂的界以及特征值比率的显式区间。涉及势的梯度和Laplacian的界给出了进一步的比较。在黎曼球面上，我们获得了单个特征值、倒数之和以及计数函数的估计。对于单位二维球面上的Kähler度量，若其为中心化（即单位法向量场积分为零），则势的Dirichlet能量可定量改进Hersch关于第一正特征值和前三个倒数特征值之和的界。
英文摘要：
We establish quantitative comparisons between the Laplace--Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.
