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El. knyga: Regularity Theory for Mean Curvature Flow

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This work is devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow. Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics. A major example is Hamilton's Ricci flow program, which has the aim of settling Thurston's geometrization conjecture, with recent major progress due to Perelman. Another important application of a curvature flow process is the resolution of the famous Penrose conjecture in general relativity by Huisken and Ilmanen. Under mean curvature flow, surfaces usually develop singularities in finite time. This work presents techniques for the study of singularities of mean curvature flow and is largely based on the work of K. Brakke, although more recent developments are incorporated. 

This work is devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow. Mean curvature flow and related flows are important tools in mathematics and mathematical physics. For example, the famous Penrose conjecture in general relativity by Huisken and Ilmanan was based on a curvature flow approach. Under mean curvature flow, surfaces usually develop singularities in finite time. This work presents techniques in the study of singularities of mean curvature flow and is largely based on the work of K. Brakke, although more recent developments will be presented as well: for example, relations to regularity theory for minimal surfaces, as in Allard's and de Giorgi's work. Graduate students and researchers in nonlinear PDEs, geometric measure theory and mathematical physics will benefit from this work.

Recenzijos

"The central theme [ in this book] is the regularity theory for mean curvature flow leading to a clear simplified proof of Brakke's main regularity theorem for this special case.... [ The] author gives a detailed account of techniques for the study of singularities and expresses the underlying ideas almost entirely in the language of differential geometry and partial differential equations.... This is a very nice book. The presentations are very clear and direct. Graduate students and researchers in differential geometry and partial differential equations will benefit from this work."



Mathematical Reviews



"For the last 20 years, the computational and theoretical study and application of generalized motion by mean curvature and more general curvature flows have had enormous impact in diverse areas of pure and applied mathematics. Klaus Ecker's new book provides an attractive, elegant, and largely self-contained introduction to the study of classical mean curvature flow, developing some fundamental ideas from minimal surface theory...all with the aim of proving a version of Brakke's regularity theorem and estimating the size of the 'singular set.' In order to limit technicalities, the discussion is basically limited to classical flows up until a first singularity develops. This makes the book very readable and suitable for students and applied mathematicians who want to gain more insight into the subtleties of the subject."



SIAM Review



"This book offers an introduction to Brakke's reuglarity theory for the mean curvature flow, incorporating many simplifications of the arguments, which have been found during the last decades." ---Monatshefte für Mathematik



"The book...is a short and very readable account on recent results obained about the structure of singularities. [ I]t is definitely an intersting purchase if one wants to gain some technical insight in related nonlinearevolution problems such as the harmonic map heat flow or Hamilton's Ricci flow for metrics." ---Mathematical Society

Daugiau informacijos

Springer Book Archives
Preface ix
1 Introduction 1 (6)
2 Special Solutions and Global Behaviour 7 (16)
3 Local Estimates via the Maximum Principle 23 (24)
4 Integral Estimates and Monotonicity Formulas 47 (34)
5 Regularity Theory at the First Singular Time 81 (28)
A Geometry of Hypersurfaces 109(10)
B Derivation of the Evolution Equations 119 (4)
C Background on Geometric Measure Theory 123(4)
D Local Results for Minimal Hypersurfaces 127(12)
E Remarks on Brakke's Clearing Out Lemma 139 (6)
F Local Monotonicity in Closed Form 145 (8)
Bibliography 153 (6)
Index 159


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