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The regularity of limit spaces of Riemannian manifolds with L^p curvaturebounds, $p > n/2$, is investigated under no apriori non-collapsing assumption.A regular subset, defined by a local volume growth condition for a limitmeasure, is shown to carry the structure of a Riemannian manifold. Oneconsequence of this is a compactness theorem for Riemannian manifolds with$L^p$ curvature bounds and an a priori volume growth assumption in the pointedCheeger-Gromov topology. A different notion of convergence is also studied,which replaces the exhaustion by balls in the pointed Cheeger-Gromov topologywith an exhaustion by volume non-collapsed regions. Assuming in addition alower bound on the Ricci curvature, the compactness theorem is extended to thistopology. Moreover, we study how a convergent sequence of manifolds disconnectstopologically 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 unionof an incomplete Riemannian surface and 1-dimensional length spaces.

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