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 index
Graduate students and researchers in nonlinear PDEs, differential geometry, geometric measure theory and mathematical physics will benefit from this work. TOC:Preface.- Introduction.- Special Solutions and Global Behaviour.- Local Estimates via the Maximum Principle.- Integral Estimates and Monotonicity Formulas.- Regularity Theory at the First Singular Time.- A Geometry of Hypersurfaces.- Derivation of the Evolution Equations.- Background on Geometric Measure Theory.- Local Results for Minimal Hypersurfaces.- Remarks on Brakke's Clearing Out Lemma.- Local Monotonicity in Closed Form.- Bibliography.- Index.
