{"paper":{"title":"Contracting axially symmetric hypersurfaces by powers of the $\\sigma_k$-curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Haizhong Li, Jing Wu, Xianfeng Wang","submitted_at":"2019-05-14T12:58:32Z","abstract_excerpt":"In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\\mathbb{R}^{n+1}$ and $\\mathbb{S}^{n+1}$ by $\\sigma_k^\\alpha$, where $\\sigma_k$ is the $k$-th elementary symmetric function of the principal curvatures and $\\alpha\\ge 1/k$. We prove that for any $n\\geq3$ and any fixed $k$ with $1\\leq k\\leq n$, there exists a constant $c(n,k)>1/k$ such that that if $\\alpha$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.05571","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"}