Diameter Controls and Smooth Convergence away from Singular Sets

TL;DR

This paper establishes conditions under which a sequence of Riemannian metrics converges to a limit space, focusing on diameter controls and smooth convergence away from singular sets, extending previous results to more general singularities.

Contribution

It generalizes prior convergence results by replacing codimension two singularity assumptions with Hausdorff measure conditions and diameter bounds, and introduces new convergence criteria involving linear contractibility.

Findings

Gromov-Hausdorff limit exists under specified conditions.

Convergence can be characterized via Sormani-Wenger Intrinsic Flat distance.

Examples illustrate the necessity of hypotheses.

Abstract

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$, and if there exists connected precompact exhaustion, $W_j$, of $M^n \setminus S$ satisfying $\diam_{g_i}(M^n) \le D_0 $, $\vol_{g_i}(\partial W_j) \le A_0 $ and $\vol_{g_i}(M^n \setminus W_j) \le V_j where \lim_{j\to\infty}V_j=0 $ then the Gromov-Hausdorff limit exists and agrees with the metric completion of $(M^n \setminus S, g_\infty)$. Recall that in the prior work with Sormani the same conclusion is reached but the singular set is assumed to be a submanifold of codimension two. We have a second main theorem in which the Hausdorff measure condition on $S$ is replaced by diameter estimates on the connected components of the boundary of the exhaustion, $\partial W_j$. In addition, we show that the uniform lower Ricci curvature bounds in these theorems can be replaced by the existence of a uniform linear contractibility function. If this condition is removed altogether, then we prove that $\lim_{j\to \infty} d_{\mathcal{F}}(M_j', N')=0$, in which $M_j'$ and $N'$ are the settled completions of $(M, g_j)$ and $(M_\infty\setminus S, g_\infty)$ respectively and $d_{\mathcal{F}}$ is the Sormani-Wenger Intrinsic Flat distance. We present examples demonstrating the necessity of many of the hypotheses in our theorems.