The avoidance principle for noncompact hypersurfaces moving by mean curvature flow

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI:10.1007/s00526-024-02725-5

Brian White

引用次数: 0

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.

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以平均曲率流运动的非紧凑超曲面的回避原理

考虑一对光滑的、可能非紧凑的、适当浸没的超曲面，它们通过平均曲率流（或更广义地说，一对弱集流）运动。我们证明，如果环境空间是欧几里得空间，如果两个曲面之间的距离最初不为零，那么这两个曲面在随后的所有时间里都保持不相交。当环境空间是具有非零注入半径的完整黎曼流形时，只要（环境空间的）曲率张量及其所有导数都是有界的，我们就能证明同样的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.