{"paper":{"title":"On the monotonicity of spatial critical points evolving under curvature-driven flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.soft","math.DG"],"primary_cat":"math.AP","authors_text":"G. Domokos","submitted_at":"2013-08-22T07:18:43Z","abstract_excerpt":"We describe the variation of the number $N(t)$ of spatial critical points of smooth curves (defined as a scalar distance $r$ from a fixed origin $O$) evolving under curvature-driven flows. In the latter, the speed $v$ in the direction of the surface normal may only depend on the curvature $\\kappa$. Under the assumption that only generic saddle-node bifurcations occur, we show that $N(t)$ will decrease if the partial derivative $v_{\\kappa}$ is positive and increase if it is negative (Theorem 1). Justification for the genericity assumption is provided in Section 5. For surfaces embedded in 3D, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4779","kind":"arxiv","version":4},"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"}