{"paper":{"title":"Volume growth, eigenvalue and compactness for self-shrinkers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Qi Ding, Y. L. Xin","submitted_at":"2011-01-07T11:51:29Z","abstract_excerpt":"In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \\cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\\ir{3}$ obtained by Colding-Minicozzi under weaker conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1411","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"}