Regularity Theory for Mean Curvature Flow: (57 Progress in Nonlinear Differential Equations and Their Applications)

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.Key features and topics of this self-contained, systematic exposition:* A detailed account of techniques (including some of Brakke's original ones) leading to a proof of Brakke's main regularity theorem* Preliminary material begins with the concept of mean curvature flow, illustrated with important examples and special solutions including a detailed discussion of homethetic solutions* Local pointwise estimates on geometric quantities for smooth solutions of mean curvature flow are derived in a streamlined presentation* Rescaling methods, monotonicity formulas, and mean value inequalities are presented* Two local regularity theorems and an estimate of the singular set are established* Definitions and facts for hypersurfaces in Euclidean space, used throughout the text, are listed in an appendix, along with some background on geometric measure theory and minimal surface theory* Good bibliography and indexGraduate students and researchers in nonlinear PDEs, differential geometry, geometric measure theory and mathematical physics will benefit from this work.

Table of Contents:
1 Introduction.- 2 Special Solutions and Global Behaviour.- 3 Local Estimates via the Maximum Principle.- 4 Integral Estimates and Monotonicity Formulas.- 5 Regularity Theory at the First Singular Time.- A Geometry of Hypersurfaces.- B Derivation of the Evolution Equations.- C Background on Geometric Measure Theory.- D Local Results for Minimal Hypersurfaces.- E Remarks on Brakke¡¯s Clearing Out Lemma.- F Local Monotonicity in Closed Form.

Review :
"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 fur 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 nonlinear evolution problems such as the harmonic map heat flow or Hamilton's Ricci flow for metrics." ---Mathematical Society