Format:
Online-Ressource
,
v.: digital
Edition:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
9783034801454
Series Statement:
Progress in Mathematics 290
Content:
Carlo Mantegazza
Content:
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
Note:
Includes bibliographical references and index
,
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
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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
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5.1.1 Embedded Curves5.1.2 General Curves; 5.2 Hypersurfaces; 5.2.1 Entire Graphs; 5.2.2 Convex Hypersurfaces; 5.2.3 Embedded Mean Convex Hypersurfaces; 5.2.4 Two-Convex Hypersurfaces; 5.2.5 General Hypersurfaces; 5.3 Mean Curvature Flow with Surgeries; 5.4 Some Problems and Research Directions; 5.4.1 Motion of Noncompact Hypersurfaces; 5.4.2 Motion of Hypersurfaces with Boundary; 5.4.3 Higher Codimension; 5.4.4 Evolutions by Different Functions of the Curvature; 5.4.5 Weak Solutions; Appendix A: Quasilinear Parabolic Equations on Manifolds; A.1 The Linear Case
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A.2 Regularity in the Linear CaseA.3 The General Case; 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;
,
Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 9783034801447
Language:
English
Subjects:
Mathematics
Keywords:
Mittlere Krümmung
;
Fluss
DOI:
10.1007/978-3-0348-0145-4
URL:
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