Level set approach to mean curvature flow in arbitrary codimension

Luigi Ambrosio, Halil Mete Soner

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160 Scopus citations

Abstract

We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of codimension-k in Rd by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d - k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution is easily established by using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and its connection to the level set solutions is also provided.

Original languageEnglish (US)
Pages (from-to)693-737
Number of pages45
JournalJournal of Differential Geometry
Volume43
Issue number4
DOIs
StatePublished - 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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