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Brakke's Mean Curvature Flow - An Introduction (Paperback, 1st ed. 2019)
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Brakke's Mean Curvature Flow - An Introduction (Paperback, 1st ed. 2019)
Series: SpringerBriefs in Mathematics
Expected to ship within 10 - 15 working days
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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.
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