{"paper":{"title":"The curve shortening flow with parallel 1-form","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hengyu Zhou","submitted_at":"2012-12-21T16:39:34Z","abstract_excerpt":"Let $M$ be a closed Riemannian manifold with a parallel 1-form $\\Omega$. We prove two theorems about the curve shortening flow in $M$. One is that the {\\csf} $\\ct$ in $M$ exists for all $t$ in $[0, \\infty)$, if it satisfies $\\Omega(T)\\geq 0$ on the initial curve $\\co$. Here $T$ is the unit tangent vector on $\\co$. The other one is about the convergence. It says that in a closed {\\Rm} $\\tilde{M}$, assume the curve shortening flow $\\ct$ exists for all $t\\in[0,\\infty)$ and its length converges to a positive limit, then $ \\lim\\limits_{t\\rightarrow\\infty}max_{\\ct}|\\nabla^{m}A|^{2}=0$ for all $m=0,1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5515","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"}