Mean Curvature Flow and Isoperimetric Inequalities

  • Manuel Ritoré
  • Carlo Sinestrari

Part of the Advanced Courses in Mathematics — CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Formation of Singularities in the Mean Curvature Flow

    1. Front Matter
      Pages 1-1
    2. Manuel Ritoré, Carlo Sinestrari
      Pages 3-5
    3. Manuel Ritoré, Carlo Sinestrari
      Pages 5-9
    4. Manuel Ritoré, Carlo Sinestrari
      Pages 9-10
    5. Manuel Ritoré, Carlo Sinestrari
      Pages 10-16
    6. Manuel Ritoré, Carlo Sinestrari
      Pages 16-19
    7. Manuel Ritoré, Carlo Sinestrari
      Pages 20-23
    8. Manuel Ritoré, Carlo Sinestrari
      Pages 23-25
    9. Manuel Ritoré, Carlo Sinestrari
      Pages 25-28
    10. Manuel Ritoré, Carlo Sinestrari
      Pages 28-32
    11. Manuel Ritoré, Carlo Sinestrari
      Pages 32-35
    12. Manuel Ritoré, Carlo Sinestrari
      Pages 36-38
  3. Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry

    1. Front Matter
      Pages 45-52
    2. Manuel Ritoré, Carlo Sinestrari
      Pages 53-67
    3. Manuel Ritoré, Carlo Sinestrari
      Pages 69-83
    4. Manuel Ritoré, Carlo Sinestrari
      Pages 85-97
    5. Manuel Ritoré, Carlo Sinestrari
      Pages 99-103
  4. Back Matter
    Pages 105-113

About this book

Introduction

Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.

Keywords

Mean curvature Minimal surface Ricci flow curvature manifold

Authors and affiliations

  • Manuel Ritoré
    • 1
  • Carlo Sinestrari
    • 2
  1. 1.Departamento de Geometría y Topología Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata” Via della Ricerca ScientificaRomaItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0346-0213-6
  • Copyright Information Birkhäuser Basel 2010
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0346-0212-9
  • Online ISBN 978-3-0346-0213-6
  • About this book