Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

2020/07/14 by Laiyuan Gao, Gao, Laiyuan
Mathematics · #35K15 #51M05 #53A07 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.07342

openalex publication_date 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₀, \widetildeX be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve X₀ into \widetildeX. It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve \widetildeX as time tends to infinity.

Citations

• An isoperimetric inequality with applications to curve shortening

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