Smooth Convergence Away from Singular Sets

TL;DR

This paper investigates the convergence of Riemannian metrics away from singular sets, establishing conditions for Gromov-Hausdorff convergence to metric completions and introducing a new hemispherical embedding method.

Contribution

It introduces a novel hemispherical embedding technique and provides new theorems describing convergence behavior of metrics with singularities, along with corrected statements following a counterexample.

Findings

Established conditions for Gromov-Hausdorff convergence to metric completions.

Developed the hemispherical embedding method for explicit distance estimates.

Corrected previous theorems based on counterexamples.

Abstract

We consider sequences of metrics, $g_j$, on a 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 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 Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains. Seven years after the publication of this paper in CAG, Brian Allen discovered a counter example to the published statement of Theorem 1.3. Note that Theorem 4.6 (which is the key theorem cited in other papers) remains correct. We have added an hypothesis to correct the statement of Theorem 1.3 and its consequences. This v4 includes corrections in blue, an erratum at the end of the introduction, and Brian Allen's example in an appendix. An erratum is also being sent to the journal.