Manifolds with small curvature concentration

TL;DR

This paper develops new tools for analyzing manifolds with small curvature concentration, enabling the construction of Ricci flows and revealing their topological structure under certain curvature and entropy conditions.

Contribution

It introduces methods to construct distance-like functions with integral Hessian bounds on such manifolds and applies these to prove topological Euclideanness.

Findings

Manifolds with Ricci lower bound and small curvature concentration are topologically Euclidean.

Constructed Ricci flows on manifolds with unbounded curvature.

Demonstrated the geometric structure of manifolds without bounded curvature assumption.

Abstract

In this work, we construct distance like functions with integral hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors $n$-regular and small curvature concentration are topologically Euclidean.