[Distinguished Lecture] Rectifiability and Mass Bounds for Singular Sets in Nonc
发布时间：2018年06月30日
浏览次数：6688
发布者: Ying Hao
主讲人： Jeff Cheeger (New York University)
活动时间： 从 2018-07-03 14:00 到 15:00
场地： 北京国际数学研究中心，镜春园78号院（怀新园）77201室
Abstract:
We will talk about recent joint work with Aaron Naber and Wenshuai Jiang. Let $(M^n_i,p_i)\stackrel{d_GH}{\longrightarrow} (X^n,x)$ denote a pointed Gromov-Hausdorff limit space with ${\rm Vol}(B_1(p_i))>{\rm v}$ and ${\rm Ric}_{M^n_i}\geq -(n-1)$. It is known that the singular set $S\subset X^n$ can be dense. However, from our work with T. Colding, it is known that for all $\epsilon>0$, there is a decomposition, $X^n=\mathcal R_\epsilon\cup S_\epsilon$, where $S_\epsilon$ is closed, $S=\bigcup_\epsilon\, S_\epsilon$, the Hausdorff dimension of $S_\epsilon$ is $\leq n-2$ and $\mathcal R_\epsilon$ is $\theta(\epsilon)$-bi-H\"older equivalent to a smooth riemannian manifold, where $\theta(\epsilon)\to 1$ as $\epsilon\to 0$. Our main theorem states that $S_\epsilon$ (and hence, $S$) is rectifiable and there is a Hausdorff measure bound $\mathcal H^{n-2}(S_\epsilon\cap B_1(p_i))\leq c(n,{\rm v},\epsilon)$. These statements are consequences of stronger effective bounds on the so called "quantitative stratification" and still more general statements on "neck regions".
Introduction of the speaker:
Professor Jeff Cheeger has been a member of the National Academy of Sciences of the United States since 1997. He received the Oswald Veblen Prize in Geometry from the American Mathematical Society in 2001. His main interests are differential geometry and its connections with topology and analysis.
We will talk about recent joint work with Aaron Naber and Wenshuai Jiang. Let $(M^n_i,p_i)\stackrel{d_GH}{\longrightarrow} (X^n,x)$ denote a pointed Gromov-Hausdorff limit space with ${\rm Vol}(B_1(p_i))>{\rm v}$ and ${\rm Ric}_{M^n_i}\geq -(n-1)$. It is known that the singular set $S\subset X^n$ can be dense. However, from our work with T. Colding, it is known that for all $\epsilon>0$, there is a decomposition, $X^n=\mathcal R_\epsilon\cup S_\epsilon$, where $S_\epsilon$ is closed, $S=\bigcup_\epsilon\, S_\epsilon$, the Hausdorff dimension of $S_\epsilon$ is $\leq n-2$ and $\mathcal R_\epsilon$ is $\theta(\epsilon)$-bi-H\"older equivalent to a smooth riemannian manifold, where $\theta(\epsilon)\to 1$ as $\epsilon\to 0$. Our main theorem states that $S_\epsilon$ (and hence, $S$) is rectifiable and there is a Hausdorff measure bound $\mathcal H^{n-2}(S_\epsilon\cap B_1(p_i))\leq c(n,{\rm v},\epsilon)$. These statements are consequences of stronger effective bounds on the so called "quantitative stratification" and still more general statements on "neck regions".
Introduction of the speaker:
Professor Jeff Cheeger has been a member of the National Academy of Sciences of the United States since 1997. He received the Oswald Veblen Prize in Geometry from the American Mathematical Society in 2001. His main interests are differential geometry and its connections with topology and analysis.
