This book explains the notion of Brakke's mean curvature flow and its
existence and regularity theories without assuming familiarity with
geometric measure theory. The focus of study is a time-parameterized
family of k-dimensional surfaces in the n-dimensional Euclidean
space (1 k n). The family is the mean curvature flow if the velocity of
motion of surfaces is given by the mean curvature at each point and
time. It is one of the simplest and most important geometric evolution
problems with a strong connection to minimal surface theory. In fact,
equilibrium of mean curvature flow corresponds precisely to minimal
surface. Brakke's mean curvature flow was first introduced in 1978 as a
mathematical model describing the motion of grain boundaries in an
annealing pure metal. The grain boundaries move by the mean curvature
flow while retaining singularities such as triple junction points. By
using a notion of generalized surface called a varifold from geometric
measure theory which allows the presence of singularities, Brakke
successfully gave it a definition and presented its existence and
regularity theories. Recently, the author provided a complete proof of
Brakke's existence and regularity theorems, which form the content of
the latter half of the book. The regularity theorem is also a natural
generalization of Allard's regularity theorem, which is a fundamental
regularity result for minimal surfaces and for surfaces with bounded
mean curvature. By carefully presenting a minimal amount of mathematical
tools, often only with intuitive explanation, this book serves as a good
starting point for the study of this fascinating object as well as a
comprehensive introduction to other important notions from geometric
measure theory.
