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 (GH) sense to the metric completion, (M_∞, d_∞), of (M S, g_∞). 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 GH and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains.
展开▼
