On limit spaces of Riemannian manifolds with volume and integral curvature bounds

TL;DR

This paper studies the structure and compactness of limit spaces of Riemannian manifolds with L^p curvature bounds, establishing regularity, convergence, and topological properties under volume and curvature conditions.

Contribution

It introduces new regularity results and compactness theorems for limit spaces of manifolds with L^p curvature bounds without non-collapsing assumptions.

Findings

Regular subset of limit spaces has Riemannian manifold structure.

Established compactness in Cheeger-Gromov topology under volume and curvature bounds.

Described the detailed structure of 2D limit spaces as unions of surfaces and 1D spaces.

Abstract

The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an a priori volume growth assumption in the pointed Cheeger-Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger-Gromov topology with an exhaustion by volume non-collapsed regions. Assuming in addition a lower bound on the Ricci curvature, the compactness theorem is extended to this topology. Moreover, we study how a convergent sequence of manifolds disconnects topologically in the limit. In two dimensions, building on results of Shioya, the structure of limit spaces is described in detail: it is seen to be a union of an incomplete Riemannian surface and 1-dimensional length spaces.