## [Distinguished Lecture] Rectifiability and Mass Bounds for Singular Sets in Noncollapsed Gromov-Hausdorff Limit Spaces With Ricci Curvature Bounded Below

**Speaker(s): ** Jeff Cheeger (New York University)

**Time: ** 14:00-15:00 July 3, 2018

**Venue: ** Room 77201, Jingchunyuan 78, BICMR

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.