Regularity Theory for Mean Curvature Flow [Hardcover]

* 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.

 

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.

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 regullsB