January 2021 $L^2$ curvature bounds on manifolds with bounded Ricci curvature
Wenshuai Jiang, Aaron Naber
Author Affiliations +
Ann. of Math. (2) 193(1): 107-222 (January 2021). DOI: 10.4007/annals.2021.193.1.2

Abstract

Consider a Riemannian manifold with bounded Ricci curvature $|\mathrm{Ric}|\leq n-1$ and the noncollapsing lower volume bound $\mathrm{Vol}(B_1(p))>\mathrm{v}>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $⨏_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx \lt C(n,\mathrm{v})$,which proves the $L^2$ conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if $(M^n_j,d_j,p_j) \longrightarrow (X,d,p)$ is a $\mathrm{GH}$-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set $\mathcal{S}(X)$ is $n-4$ rectifiable with the uniform Hausdorff measure estimates $H^{n-4}(\mathcal{S}(X)\cap B_1) \lt C(n,\mathrm{v})$ which, in particular, proves the $n-4$-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for $n-4$ a.e.\ $x\in \mathcal{S}(X)$, the tangent cone of $X$ at $x$ is unique and isometric to $\mathbb{R}^{n-4}\times C(S^3/\Gamma_x)$ for some $\Gamma_x\subseteq O(4)$ that acts freely away from the origin.

Citation

Download Citation

Wenshuai Jiang. Aaron Naber. "$L^2$ curvature bounds on manifolds with bounded Ricci curvature." Ann. of Math. (2) 193 (1) 107 - 222, January 2021. https://doi.org/10.4007/annals.2021.193.1.2

Information

Published: January 2021
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2021.193.1.2

Subjects:
Primary: 35A21 , 53B20

Keywords: curvature , Ricci , singularity , stratification

Rights: Copyright © 2021 Department of Mathematics, Princeton University

JOURNAL ARTICLE
116 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.193 • No. 1 • January 2021
Back to Top