The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow

2024/01/25 by Brian White, White, Brian · 3 citations
Mathematics · #53E10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2401.13966

openalex publication_date 2024/01/25 · openalex created_date 2024/01/27 · openalex updated_date 2026/07/28

Abstract

Consider a pair of smooth, possibly noncompact, properly immersed hypersurfaces moving by mean curvature flow, or, more generally, a pair of weak set flows. We prove that if the ambient space is Euclidean space and if the distance between the two surfaces is initially nonzero, then the surfaces remain disjoint at all subsequent times. We prove the same result when the ambient space is a complete Riemannian manifold of nonzero injectivity radius, provided the curvature tensor (of the ambient space) and all its derivatives are bounded.

Cited by

• Equivalence of weak solution concepts for mean curvature flow
• An Ancient Stacked Pancake Solution to Mean Curvature Flow
• An Intersection Principle for Mean Curvature Flow

Related

• An Intersection Principle for Mean Curvature Flow
• Non-collapsing in mean-convex mean curvature flow
• Weak solutions to mean curvature flow respecting obstacles I: the\n graphical case
• Non-preserved curvature conditions under constrained mean curvature flows
• General curvature flow without singularities