Communications in Analysis and Geometry

Volume 21 (2013)

Number 1

Smooth convergence away from singular sets

Pages: 39 – 104



Sajjad Lakzian (Graduate Center, City University of New York)

Christina Sormani (Graduate Center and Lehman College, City University of New York)


We consider sequences of metrics, $g_j$, on a compact Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $S\subset M$, to a metric, $g_\infty$, on $M\setminus S$. We prove theorems which describe when $M_j=(M, g_j)$ converge in the Gromov–Hausdorff (GH) sense to the metric completion, $(M_\infty,d_\infty)$, of $(M\setminus S, g_\infty)$. 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.

Published 9 April 2013