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

Abstract

The regularity of limit spaces of Riemannian manifolds with $L^p$ curvature bounds, $p \gt n/2$, is investigated under no apriori noncollapsing 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.