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Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

  • Textbook
  • © 2013

Overview

Authors:
  1. Giovanni Bellettini

    1. Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italia

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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

Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 12)

Part of the book sub series: Lecture Notes (Scuola Normale Superiore) (LNSNS)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. Signed distance from a smooth boundary

    • Giovanni Bellettini
    Pages 1-22

  3. First variations

    • Giovanni Bellettini
    Pages 23-35

  4. Smooth flows

    • Giovanni Bellettini
    Pages 37-58

  5. Huisken’s monotonicity formula

    • Giovanni Bellettini
    Pages 59-68

  6. Inclusion principle

    • Giovanni Bellettini
    Pages 69-84

  7. Extension of the evolution equation to a neighbourhood

    • Giovanni Bellettini
    Pages 85-101

  8. Local well-posedness: the approach of Evans and Spruck

    • Giovanni Bellettini
    Pages 103-126

  9. Grayson’s example

    • Giovanni Bellettini
    Pages 127-134

  10. De Giorgi’s barriers

    • Giovanni Bellettini
    Pages 135-163

  11. Inner and outer regularizations

    • Giovanni Bellettini
    Pages 165-174

  12. An example of fattening

    • Giovanni Bellettini
    Pages 175-185

  13. Ilmanen’s interposition lemma

    • Giovanni Bellettini
    Pages 187-205

  14. The avoidance principle

    • Giovanni Bellettini
    Pages 207-215

  15. Comparison between barriers and a generalized evolution

    • Giovanni Bellettini
    Pages 217-227

  16. One-dimensional analysis related to a reaction-diffusion equation

    • Giovanni Bellettini
    Pages 229-247

  17. Parabolic singular perturbations: formal matched asymptotics

    • Giovanni Bellettini
    Pages 249-262

  18. Parabolic singular perturbations: convergence and error estimate

    • Giovanni Bellettini
    Pages 263-295

  19. Back Matter

    Pages 297-329
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Keywords

About this book

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.

Authors and Affiliations

  • Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italia

    Giovanni Bellettini

Bibliographic Information

  • Book Title: Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

  • Authors: Giovanni Bellettini

  • Series Title: Publications of the Scuola Normale Superiore

  • DOI: https://doi.org/10.1007/978-88-7642-429-8

  • Publisher: Edizioni della Normale Pisa

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Edizioni della Normale 2013

  • Softcover ISBN: 978-88-7642-428-1Published: 16 January 2014

  • eBook ISBN: 978-88-7642-429-8Published: 13 May 2014

  • Series ISSN: 2239-1460

  • Series E-ISSN: 2532-1668

  • Edition Number: 1

  • Number of Pages: XVIII, 329

  • Topics: Geometry

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