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Lectures on mean curvature flow of surfaces

Robert Haslhofer

Abstract

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and nonuniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, weak solutions, and blowup analysis that are of great importance in the analysis of a wide range of partial differential equations. These lecture notes are from summer schools at UT Austin and CRM Montreal, and also contain a detailed discussion of open problems and conjectures.

Lectures on mean curvature flow of surfaces

Abstract

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and nonuniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, weak solutions, and blowup analysis that are of great importance in the analysis of a wide range of partial differential equations. These lecture notes are from summer schools at UT Austin and CRM Montreal, and also contain a detailed discussion of open problems and conjectures.

Paper Structure

This paper contains 6 sections, 19 theorems, 62 equations, 1 figure.

Table of Contents

  1. Overview and basic properties
  2. Monotonicity formula and epsilon-regularity
  3. Noncollapsing, curvature and convexity estimate
  4. Weak solutions
  5. Flow through singularities
  6. Open problems

Key Result

Theorem 1.6

Let $M_0\subset \mathbb R^{3}$ be a closed embedded surface. If $M_0$ is convex, then the mean curvature flow $\{M_t\}_{t\in[0,T)}$ starting at $M_0$ converges to a round point.

Figures (1)

  • Figure 1: Cylinder, bowl and ancient oval.

Theorems & Definitions (39)

  • Example 1.2: Shrinking sphere and cylinder
  • Theorem 1.6: Huisken's convergence theorem
  • Example 1.7: Neckpinch singularity
  • Proposition 1.9: Evolution equations for geometric quantities
  • Corollary 1.14: mean-convexity and convexity
  • Theorem 2.2: Huisken's monotonicity formula
  • proof : Proof of Theorem \ref{['thm_huisken_mon']}
  • Theorem 2.14: epsilon-regularity
  • proof
  • Definition 3.1: noncollapsing
  • ...and 29 more