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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 written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.
List of contents
Foreword.- Chapter 1. Definition and Short Time Existence.- Chapter 2. Evolution of Geometric Quantities.- Chapter 3. Monotonicity Formula and Type I Singularities.- Chapter 4. Type II Singularities.- Chapter 5. Conclusions and Research Directions.- Appendix A. Quasilinear Parabolic Equations on Manifolds.- Appendix B. Interior Estimates of Ecker and Huisken.- Appendix C. Hamilton's Maximum Principle for Tensors.- Appendix D. Hamilton's Matrix Li-Yau-Harnack Inequality in Rn.- Appendix E. Abresch and Langer Classification of Homothetically Shrinking Closed Curves.- Appendix F. Important Results without Proof in the Book.- Bibliography.- Index.
Summary
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 written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.
Additional text
From the book reviews:
“This award-winning monograph provides an introduction to the topic of mean curvature flow of hypersurfaces in Euclidean space for the advanced student and the researcher … . It reorganizes material scattered throughout the literature within the last 25 years, thereby mainly concentrating on the classical parametric approach due to R. Hamilton and G. Huisken.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)
“In the book under review the author mainly discusses some classical results on mean curvature flow of hypersurfaces. … Specifically, the author also gives some recent conclusions, some references to open problems and research directions. The book is not only suitable for beginners but also for researchers.” (Shouwen Fang, Mathematical Reviews, January, 2013)
“This book gives an introduction to the topic of mean curvature flows of hypersurfaces in Euclidean spaces. … It is written in the style of lecture notes and provides a detailed discussion of the classical parametric approach by R. Hamilton and G. Huisken, as well as the results by other authors scattered over the literature of the last 25 years. … The book finishes with 6 appendices.” (Boris S. Kruglikov, Zentralblatt MATH, Vol. 1230, 2012)
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From the book reviews:
"This award-winning monograph provides an introduction to the topic of mean curvature flow of hypersurfaces in Euclidean space for the advanced student and the researcher ... . It reorganizes material scattered throughout the literature within the last 25 years, thereby mainly concentrating on the classical parametric approach due to R. Hamilton and G. Huisken." (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)
"In the book under review the author mainly discusses some classical results on mean curvature flow of hypersurfaces. ... Specifically, the author also gives some recent conclusions, some references to open problems and research directions. The book is not only suitable for beginners but also for researchers." (Shouwen Fang, Mathematical Reviews, January, 2013)
"This book gives an introduction to the topic of mean curvature flows of hypersurfaces in Euclidean spaces. ... It is written in the style of lecture notes and provides a detailed discussion of the classical parametric approach by R. Hamilton and G. Huisken, as well as the results by other authors scattered over the literature of the last 25 years. ... The book finishes with 6 appendices." (Boris S. Kruglikov, Zentralblatt MATH, Vol. 1230, 2012)
