'Mean curvature flow' is a term that is used to describe the evolution of a hyper surface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton's theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals $\pi$, the curve tends to the unit circle. In this book, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior of mean curvature flows in higher dimensions.Among other topics, he considers in detail Huisken's theorem (a generalization of Gage-Hamilton's theorem to higher dimension), evolution of non-convex curves and hypersurfaces, and the classification of singularities of the mean curvature flow. Because of the importance of the mean curvature flow and its numerous applications in differential geometry and partial differential equations, as well as in engineering, chemistry, and biology, this book can be useful to graduate students and researchers working in these areas. The book would also make a nice supplementary text for an advanced course in differential geometry. Prerequisites include basic differential geometry, partial differential equations, and related applications.
The curve shortening flow for convex curves The short time existence and the evolution equation of curvatures Contraction of convex hypersurfaces Monotonicity and self-similar solutions Evolution of embedded curves or surfaces (I) Evolution of embedded curves and surfaces (II) Evolution of embedded curves and surfaces (III) Convexity estimates for mean convex surfaces Li-Yau estimates and type II singularities The mean curvature flow in Riemannian manifolds Contracting convex hypersurfaces in Riemannian manifolds Definition of center of mass for isolated gravitating systems References Index.