L2 curvature bounds on manifolds with bounded Ricci curvature

Wenshuai Jiang; Aaron Naber

doi:10.4007/annals.2021.193.1.2

L² curvature bounds on manifolds with bounded Ricci curvature

Wenshuai Jiang^*, Aaron Naber

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

24 Scopus citations

Abstract

Consider a Riemannian manifold with bounded Ricci curvature [Ric] ≤ n - 1 and the noncollapsing lower volume bound Vol(B1(p)) > v > 0. The first main result of this paper is to prove that we have the L2 curvature bound R{B1(p)jRmj2(x) dx < C(n; v), which proves the L2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, (Formula Presented) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates (Formula Presented) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for (Formula Presented) the tangent cone of X at x is unique and isometric to (Formula Presented) that acts freely away from the origin.

Original language	English (US)
Pages (from-to)	107-222
Number of pages	116
Journal	Annals of Mathematics
Volume	193
Issue number	1
DOIs	https://doi.org/10.4007/annals.2021.193.1.2
State	Published - Jan 2021

Funding

The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259. Acknowlegments. The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259.

Keywords

Curvature
Ricci
Singularity
Stratification

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.4007/annals.2021.193.1.2

Cite this

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title = "L2 curvature bounds on manifolds with bounded Ricci curvature",

abstract = "Consider a Riemannian manifold with bounded Ricci curvature [Ric] ≤ n - 1 and the noncollapsing lower volume bound Vol(B1(p)) > v > 0. The first main result of this paper is to prove that we have the L2 curvature bound R{B1(p)jRmj2(x) dx < C(n; v), which proves the L2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, (Formula Presented) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates (Formula Presented) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for (Formula Presented) the tangent cone of X at x is unique and isometric to (Formula Presented) that acts freely away from the origin.",

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author = "Wenshuai Jiang and Aaron Naber",

note = "Funding Information: The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259. Funding Information: Acknowlegments. The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259. Publisher Copyright: {\textcopyright} 2021. Department of Mathematics, Princeton University.",

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AU - Naber, Aaron

N1 - Funding Information: The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259. Funding Information: Acknowlegments. The first author would like to thank his advisor Gang Tian for constant encouragement and for useful conversations during this work. Partial work was done while the first author was visiting the Mathematics Department at Northwestern University, and he would like to thank the department for its hospitality and for providing a good academic environment. The first author was partially supported by NSFC (No. 11701507). The second author would like to thank NSF for its support under grant DMS-1406259. Publisher Copyright: © 2021. Department of Mathematics, Princeton University.

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N2 - Consider a Riemannian manifold with bounded Ricci curvature [Ric] ≤ n - 1 and the noncollapsing lower volume bound Vol(B1(p)) > v > 0. The first main result of this paper is to prove that we have the L2 curvature bound R{B1(p)jRmj2(x) dx < C(n; v), which proves the L2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, (Formula Presented) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates (Formula Presented) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for (Formula Presented) the tangent cone of X at x is unique and isometric to (Formula Presented) that acts freely away from the origin.

AB - Consider a Riemannian manifold with bounded Ricci curvature [Ric] ≤ n - 1 and the noncollapsing lower volume bound Vol(B1(p)) > v > 0. The first main result of this paper is to prove that we have the L2 curvature bound R{B1(p)jRmj2(x) dx < C(n; v), which proves the L2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, (Formula Presented) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates (Formula Presented) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for (Formula Presented) the tangent cone of X at x is unique and isometric to (Formula Presented) that acts freely away from the origin.

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