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Birkhäuser - Birkhäuser Mathematics - Scuola Normale Superiore | Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

Bellettini, Giovanni

2014, Approx. 350 p.

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  • Elementary introduction to mean curvature flow, including the short time existence result, using the signed distance function
  • Detailed study of minimal barriers and their regularizations for mean curvature flow
  • An example of fattening
  • Convergence and error estimate for the parabolic Allen-Cahn equation
The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen–Cahn (or Ginsburg–Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems.

Content Level » Graduate

Keywords » fattening - mean curvature flow - parabolic Allen-Cahn equation

Related subjects » Scuola Normale Superiore

Table of contents 

Signed distance from a smooth boundary.- Mean curvature vector and second fundamental form.- First variations of volume integrals and of the perimeter.- Smooth mean curvature flows.- Huisken’s monotonicity formula.- Inclusion principle. Local well posedness: the approach of Evans–Spruck.- Grayson’s example.- De Giorgi’s barriers.- Inner and outer regularizations.- An example of fattening.- Ilmanen’s interposition lemma.- The avoidance principle.- Comparison between barriers and a generalized evolution.- Barriers and level set evolution.- Parabolic singular perturbations: formal matched asymptotics, convergence and error estimate.

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