arXiv 2306.033822023-06math.DGmath.MG

Alexandrov 空间中 $C^2$-奇异集的定量估计

Quantitative estimates on the $C^2$-singular sets in Alexandrov spaces

Nan Li

备注：: 29 pages, added an appendix

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文针对 Alexandrov 空间中的 $C^2$-奇异点，定义了非负函数 $\mathcal K(x)$ 定量度量其偏离 $C^2$ 的程度，并给出了 Hausdorff 维数与测度的定量估计，推进了 Naber 猜想。

摘要：

如果总灾难无法预防，或许仍可被控制。我们将针对度量空间中的奇点探讨这一现象。在 $n$ 维 Alexandrov 空间中，如果一个点的切锥与 $\mathbb R^n$ 等距，则称该点为正则点。实例表明，并非所有正则点都是光滑的，且远离边界的非光滑点可能具有余维 1。本文定义了一个非负函数 $\mathcal K(x)$，用于定量度量点 $x$ 偏离 $C^2$ 的程度。所谓的 $C^2$-奇异点被识别为 $\mathcal K>0$ 的集合。我们证明，对于曲率 $\ge κ$ 且 $\operatorname{Vol}\left(B_1(p)\right)\geν>0$ 的任意 $n$ 维 Alexandrov 空间 $(X,p)$，有 $\int_{B_r(p)} \mathcal K(x)\\, \operatorname d\mathcal H^{n-1}\le c(n,κ,ν)r^{n-2}$。这导出了 Hausdorff 维数估计 $\dim_\mathcal H\{\mathcal K>0\}\le n-1$，以及定量的 Hausdorff 测度估计 $\mathcal H^{n-1}\left(\{\mathcal K>ε\}\cap B_r(p)\right)\le ε^{-1}\cdot c(n,ν)r^{n-2}$。这些结果也在 Naber 关于曲率测度收敛性的猜想上取得了进展。Alexandrov 空间上的测度 $\mathcal K(x)\\, \operatorname d\mathcal H^{n-1}$ 可视为光滑流形上曲率测度 $scal \\,\operatorname d {vol}_{g}$ 的对应物。我们还证明，若 $n$ 维 Alexandrov 空间 $X_i$ Gromov-Hausdorff 收敛到一个无边光滑流形且无坍塌，则作为测度 $\mathcal K_i\\, \operatorname d\mathcal H^{n-1}\to 0$。

英文摘要：

The total disaster may be controllable if not preventable. We will explore this phenomenon for singularities in metric spaces. A point in an $n$-dimensional Alexandrov space is called regular if its tangent cone is isometric to $\mathbb R^n$. Examples show that not every regular point is smooth, and the non-smooth points, away from the boundary, can have co-dimension 1. In this paper, we define a non-negative function $\mathcal K(x)$, which quantitatively measures the extent of the point $x$ from being $C^2$. The so-called $C^2$-singular points are identified as the set where $\mathcal K>0$. We show that $\int_{B_r(p)} \mathcal K(x)\, \operatorname d\mathcal H^{n-1}\le c(n,κ,ν)r^{n-2}$ for any $n$-dimensional Alexandrov space $(X,p)$ with curv $\ge κ$ and $\operatorname{Vol}\left(B_1(p)\right)\geν>0$. This leads to the Hausdorff dimension estimate $\dim_\mathcal H\{\mathcal K>0\}\le n-1$, and the quantitative Hausdorff measure estimate $\mathcal H^{n-1}\left(\{\mathcal K>ε\}\cap B_r(p)\right)\le ε^{-1}\cdot c(n,ν)r^{n-2}$. These results also make progress on Naber's conjecture on the convergence of curvature measures. The measure $\mathcal K(x)\, \operatorname d\mathcal H^{n-1}$ on Alexandrov spaces can be viewed as the counterpart of the curvature measure $scal \,\operatorname d {vol}_{g}$ on smooth manifolds. We also show that if $n$-dimensional Alexandrov spaces $X_i$ Gromov-Hausdorff converge to a smooth manifold with no boundary without collapsing, then $\mathcal K_i\, \operatorname d\mathcal H^{n-1}\to 0$ as a measure.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文