具下Ricci界的黎曼流形的加权离散化
A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds
- Indian Institute of Science Education and Research Mohali（印度科学教育研究所穆哈利分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对具下Ricci界的紧黎曼流形，引入其ε-离散化上的加权组合拉普拉斯算子，证明其与拉普拉斯-贝尔特拉米算子的特征值一致可比，应用该结果证明测度Gromov-Hausdorff收敛下的谱稳定性，并重新得到Schoen-Wolpert-Yau不等式。
AI中文摘要：
设$(M,g)$为连通、紧的$n$维黎曼流形，满足$\text{Ric}(M,g)\geq-(n-1)\kappa g$。我们在$M$的$\boldsymbol{\varepsilon}$-离散化上引入加权组合拉普拉斯算子，并证明加权图拉普拉斯算子与拉普拉斯-贝尔特拉米算子之间的谱比较定理；更确切地说，两个算子的特征值可通过仅依赖$n,\kappa,\varepsilon$的常数实现一致可比，且该常数与内射半径无关。作为应用，我们证明了在带测度的Gromov-Hausdorff收敛下的谱稳定性；我们还利用捏合亏格2双曲曲面族上的加权离散化，重新得到Schoen-Wolpert-Yau不等式。
英文摘要：
Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)κg$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on $n,κ,\varepsilon$, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.
