{"paper":{"title":"An Eigenvalue Pinching Theorem for Compact Hypersurfaces in A Sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongwei Xu, Yingxiang Hu","submitted_at":"2015-08-27T19:11:22Z","abstract_excerpt":"In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\\phi$ into $\\S^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\\frac{1}{n}(1+\\|B\\|_q)\\leq A$. Suppose that $\\phi(M)\\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\\frac{\\pi}{2}$. We prove that there exists a positive constant $\\e$ depending on $q$, $n$, $R$ and $A$ such that if $n(1+\\|H\\|_\\infty^2)-\\e\\leq \\l_1$, then $M$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06975","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"}