On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

2023/11/27 by Maxwell Stolarski, Stolarski, Maxwell · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #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.2311.16262

openalex publication_date 2023/11/27 · openalex created_date 2023/11/30 · openalex updated_date 2026/07/28

Abstract

This paper studies singularities of mean curvature flows with integral mean curvature bounds H ∈ L^∞ L^p_loc for some p ∈ ( n, ∞]. For such flows, any tangent flow is given by the flow of a stationary cone C. When p = ∞ and C is a regular cone, we prove that the tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset U ⊆ ℝ^N with H ∈ L^∞ L^p_loc. For smooth, codimension one mean curvature flows with H ∈ L^∞ L^∞_loc, we also show that, at points where a tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt-Simon minimal surface.

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• On the rate of convergence of cylindrical singularity in mean curvature flow

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