Diameter Controls and Smooth Convergence away from Singular Sets

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 ${\lim }_{j\to \infty }{d}_{\mathcal{F}}(M_j^{\prime },N^{\prime })=0$, in which $M_j^{\prime }$ and $N^{\prime }$ 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.

36 pages, 1 Figure; revised: Application section added

References in corpus (2)

• Singular Kahler-Einstein metrics
• Continuity of Ricci Flow Through Neck Pinch Singularities

Cited by in corpus (1)

• Properties of the Intrinsic Flat Distance