2007/03/23 by Bing-Long Chen, Chen, Bing-Long, Le Yin +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/0703694
openalex publication_date 2007/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Mean curvature flow evolves isometrically immersed base manifolds M in the direction of their mean curvatures in an ambient manifold M. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.