Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

TL;DR

This paper demonstrates that Riemannian manifolds with near-Euclidean scalar curvature, entropy, and volume bounds are Gromov-Hausdorff close to Euclidean spaces, establishing a form of geometric regularity and structure under these conditions.

Contribution

It establishes that combined scalar curvature, entropy, and volume bounds imply Gromov-Hausdorff closeness and homeomorphism to Euclidean space, extending regularity results.

Findings

Unit balls are Gromov-Hausdorff close to Euclidean balls.

Spaces are bi-Hölder and bi-W^{1,p} homeomorphic to Euclidean balls.

Proves a compactness and limit space structure theorem.

Abstract

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-H\"{o}lder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.