{"paper":{"title":"Rigidity and gap results for low index properly immersed self--shrinkers in $\\mathbb{R}^{m+1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Debora Impera","submitted_at":"2014-08-15T07:14:27Z","abstract_excerpt":"In this paper we show that the only properly immersed self--shrinkers $\\Sigma$ in $\\mathbb{R}^{m+1}$ with Morse index $1$ are the hyperplanes through the origin. Moreover, we prove that if $\\Sigma$ is not a hyperplane through the origin then the index jumps and it is at least $m+2$, with equality if and only if $\\Sigma$ is a cylinder $\\mathbb{R}^{m-k}\\times \\mathbb{S}^{k}(\\sqrt{k})$ for some $1\\leq k\\leq m-1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3479","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"}