Normal curvature bounds along the mean curvature flow

2009/06/16 by Hong Huang, Huang, Hong
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.0906.2889

Abstract

Let (Mⁿ,g₀) and (Mⁿ⁺¹,g) be complete Riemannian manifolds with |∇^kRm|≤ C for k ≤ 2, and suppose there is an isometric immersion F₀: Mⁿ → Mⁿ⁺¹ with bounded second fundamental form. Let F_t: Mⁿ → Mⁿ⁺¹ (t∈ [0,T]) be a family of immersions evolving by mean curvature flow with initial data F₀ and with uniformly bounded second fundamental forms. We show that the supremum and infimum of the normal curvature of the immersions F_t vary at a bounded rate. This is an analogue of a result of Rong and Kapovitch on Ricci flow.

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