Topology and 𝜖–regularity theorems on collapsed manifolds with Ricci curvature bounds

Naber, Aaron; Zhang, Ruobing

doi:10.2140/gt.2016.20.2575

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Topology and $\epsilon$–regularity theorems on collapsed manifolds with Ricci curvature bounds

Aaron Naber and Ruobing Zhang

Geometry & Topology 20 (2016) 2575–2664

DOI: 10.2140/gt.2016.20.2575

Abstract

In this paper we discuss and prove $ϵ$ –regularity theorems for Einstein manifolds $(M^{n}, g)$ , and more generally manifolds with just bounded Ricci curvature, in the collapsed setting.

A key tool in the regularity theory of noncollapsed Einstein manifolds is the following. If $x \in M^{n}$ is such that $V ol (B_{1} (x)) > v > 0$ and that $B_{2} (x)$ is sufficiently Gromov–Hausdorff close to a cone space $B_{2} (0^{n - ℓ}, y^{*}) \subset ℝ^{n - ℓ} \times C (Y^{ℓ - 1})$ for $ℓ \leq 3$ , then in fact $| Rm | \leq 1$ on $B_{1} (x)$ . No such results are known in the collapsed setting, and in fact it is easy to see that without further assumptions such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on $B_{1} (x)$ .

More precisely, using established techniques one can see there exists $ϵ (n)$ such that if $(M^{n}, g)$ is an Einstein manifold and $B_{2} (x)$ is $ϵ$ –Gromov–Hausdorff close to ball in $B_{2} (0^{k - ℓ}, z^{*}) \subset ℝ^{k - ℓ} \times Z^{ℓ}$ , then the fibered fundamental group $Γ_{ϵ} (x) \equiv Image [π_{1} (B_{ϵ} (x)) \to π_{1} (B_{2} (x))]$ is almost nilpotent with $rank (Γ_{ϵ} (x)) \leq n - k$ . The main result of the this paper states that if $rank (Γ_{ϵ} (x)) = n - k$ is maximal, then $| Rm | \leq C$ on $B_{1} (x)$ . In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.

Keywords

epsilon regularity, Ricci curvature

Mathematical Subject Classification 2010

Primary: 53C21, 53C25, 53B21

References

Publication

Received: 5 January 2015

Revised: 4 October 2015

Accepted: 2 November 2015

Published: 7 October 2016

Proposed: John Lott

Seconded: Bruce Kleiner, Gang Tian

Authors

Aaron Naber
Department of Mathematics Northwestern University Evanston, IL 60208 United States

Ruobing Zhang
Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NJ 08544-1000 United States

Volume 20, issue 5 (2016)