Lecture Notes on Mean Curvature Flow

This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach (mainly developed by R. Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc level and can be useful for any researcher interested in a solid introduction to the technical issues of the field. All the proofs are carefully.

Lecture Notes on Mean Curvature Flow; Contents; Foreword; Further Literature; Acknowledgment; Chapter 1: Definition and Short Time Existence; 1.1 Notation and Preliminaries; 1.2 First Variation of the Area Functional; 1.3 The Mean Curvature Flow; 1.4 Examples; 1.5 Short Time Existence of the Flow; 1.6 Other Second-order Flows; Chapter 2: Evolution of Geometric Quantities; 2.1 Maximum Principle; 2.2 Comparison Principle; 2.3 Evolution of Curvature; 2.4 Consequences of Evolution Equations; 2.5 Convexity Invariance; Chapter 3: Monotonicity Formula andType I Singularities.

3.1 The Monotonicity Formula forMean Curvature Flow3.2 Type I Singularities and the Rescaling Procedure; 3.3 Analysis of Singularities; 3.4 Hypersurfaces with Nonnegative Mean Curvature; 3.5 Embedded Closed Curves in the Plane; Chapter 4: Type II Singularities; 4.1 Hamilton's Blow-up; 4.2 Hypersurfaces with Nonnegative Mean Curvature; 4.3 The Special Case of Curves; 4.4 Hamilton's Harnack Estimate for Mean Curvature Flow; 4.5 Embedded Closed Curves in the Plane; 4.5.1 An Alternative Proof of Grayson's Theorem; Chapter 5: Conclusions and Research Directions; 5.1 Curves in the Plane.

5.1.1 Embedded Curves5.1.2 Gener…