{"paper":{"title":"Diameter Controls and Smooth Convergence away from Singular Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sajjad Lakzian","submitted_at":"2012-10-03T01:59:13Z","abstract_excerpt":"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 p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0957","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}