Regularity Theory at the First Singular Time

  • Klaus Ecker
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 57)


In this chapter we will describe and prove Brakke¡¯s main regularity theorem ([B], 6.12) in the special situation where a smooth solution of mean curvature flow develops singularities for the first time. This theorem essentially states the following. If no merging of sheets on a set of positive measure occurs when the solution becomes nonsmooth for the first time, then the singular set is lower-dimensional, that is, has n-dimensional measure zero. Before we can give a precise statement of the theorem we need some notation, definitions and some terminology from geometric measure theory.


Curvature Flow Regular Point Good Point Regularity Theory Gaussian Density 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Klaus Ecker
    • 1
  1. 1.Fachbereich für Mathematik und InformatikFreie Universität BerlinBerlinGermany

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