{"paper":{"title":"Stratified convexity & concavity of gradient flows on manifolds with boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Gabriel Katz","submitted_at":"2014-06-26T15:01:13Z","abstract_excerpt":"As has been observed by Morse \\cite{Mo}, any generic vector field $v$ on a compact smooth manifold $X$ with boundary gives rise to a stratification of the boundary $\\d X$ by compact submanifolds $\\{\\d_j^\\pm X(v)\\}_{1 \\leq j \\leq \\dim(X)}$, where $\\textup{codim}(\\d_j^\\pm X(v))= j$.\n  Our main observation is that this stratification reflects the stratified convexity/concavity of the boundary $\\d X$ with respect to the $v$-flow. We study the behavior of this stratification under deformations of the vector field $v$. We also investigate the restrictions that the existence of a convex/concave trave"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6907","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"}