Abstract. We consider sequences of metrics, g j , on a compact Riemannian manifold, M, which converge smoothly on compact sets away from a singular set S ⊂ M, to a metric, g ∞ , on M \ S . We prove theorems which describe when M j = (M, g j ) converge in the Gromov-Hausdorff sense to the metric completion,To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains.

Abstract:In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD * (K, N) spaces) with non-empty one dimensional regular sets.In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdor dimension N and the class of RCD * (K, N) spaces coincide for N < (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume suitable Ricci curvature bounds throughout the flow and also assume that S−curvature vanishes along the flow. One of the key tools we use is the Bochner identity for Finsler structures proved by .2010 Mathematics Subject Classification. 35K55(primary), and 53C21(secondary).

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