Minicourse on "Mean curvature flow and related topics"

By Francisco Martín (U. Granada)

We recommend the pre-course on Preliminaries on Submanifold Theory on September 1 & 2 to new researchers on the topic.

The aim of these lectures is giving a brief idea about singularity formation, non-uniqueness and topological change under motion by mean curvature. Mean curvature flow arises as a simplified model in several physical problems where surface tension plays a role.

Contents:

  1. Mean curvature flows in Euclidean space.
    • 1.1 Introduction.
    • 1.2 Existence.
    • 1.3 Geometric evolution equations.
    • 1.4 Comparison principle.
    • 1.5 Examples.
  2. Singularities and solitons.
    • 2.1 Types of singularities.
    • 2.2 Integral estimates and monotonicity formulas.
    • 2.3 Solitons.

References.

-Ecker, K.: Regularity Theory for Mean Curvature Flow. Birkhäuser (2004).
-Mantegazza, C: Lecture Notes on Mean Curvature Flow. Birkhäuser (2011).
-Martín F. and Pérez J.: Lecture notes on Mean curvature flow and related topics (2014). To be delivered at the meeting.
-Ritoré M. and Sinestrari C.: Mean Curvature Flow and Isoperimetric Inequalities. Birkhäuser (2010).
-Smoczyk, K.: Mean Curvature Flow in higher codimension. Introduction and survey. Preprint, arXiv:1104.3222 [math.DG].