Regularity theory for mean curvature flow. (English) Zbl 1058.53054

Progress in Nonlinear Differential Equations and their Applications 57. Boston, MA: Birkhäuser (ISBN 0-8176-3243-3/hbk; 0-8176-3781-8/pbk). xiv, 165 p. (2004).
This book is devoted to the regularity theory for mean curvature flow. The mean curvature flow evolves hypersurfaces in their normal direction with speed equal to the mean curvature, it is a descent flow for the area functional. An evolving hypersurface may become non-regular in a finite time. The regularity theory for mean curvature flow consists of results regarding the set of singular times, the dimension and structure of the singular set, the asymptotic behaviour of solutions near singularities, etc. For example, for any regular compact convex initial hypersurface \(M_0\) the solution of the mean curvature flow, \(M_t\), remains regular, compact and convex until it disappears into a round point in finite time \(t_0\) (like a family of spheres shrinking to a point).
The author’s aim is to present some recent results related to Brakke’s main regularity theorem: for the special case where smooth solutions develop singularities for the first time, this theorem states that under certain additional assumptions the hypersurfaces are still smooth at the singular time except for a lower dimensional set. In order to clarify and simplify main ideas, the author replaces Brakke’s original techniques with use of geometric measure theory by more recent methods from differential geometry and PDE theory.
The covered topics include original point-wise estimates on geometric quantities (the position vector and its derivatives, the curvature and its derivatives, etc.) for hypersurfaces evolving by mean curvature, area estimates (first variation formula, area decay, mean curvature flow in integral form, mean value inequality, etc) and Huisken’s monotonicity formula, recent developments by B. White, T. Ilmanen and K. Ecker where Brakke’s theorem has been improved. The main part of the book is followed by an appendix consisting of 6 chapters, where the author presents in more detail some background, provides an account of fundamental techniques in minimal surface theory and discusses new original results. An extended bibliography rounds off the book, including among others some monographs and articles which provide alternative approaches to mean curvature flow such as the level-set approach adopted by L. C. Evans, J. Spruck, Y. Giga and S. Goto.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
35K55 Nonlinear parabolic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions