{"paper":{"title":"Comparison Theorems for Manifold with Mean Convex Boundary","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jian Ge","submitted_at":"2013-06-21T09:20:56Z","abstract_excerpt":"Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\\in \\RR$, we give a sharp estimate of the upper bound of $\\rho(x)=\\dis(x, \\partial M)$, in terms of the mean curvature bound of the boundary. When $\\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5079","kind":"arxiv","version":1},"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"}