**AMS/IP Studies in Advanced Mathematics**

Volume: 32;
2002;
150 pp;
Hardcover

MSC: Primary 53;
Secondary 35; 52; 58

**Print ISBN: 978-0-8218-3311-7
Product Code: AMSIP/32**

List Price: $51.00

AMS Member Price: $40.80

MAA Member Price: $45.90

**Electronic ISBN: 978-1-4704-3821-0
Product Code: AMSIP/32.E**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

# Lectures on Mean Curvature Flows

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*Xi-Ping Zhu*

A co-publication of the AMS and International Press of Boston

“Mean curvature flow” is a term that is used to describe the
evolution of a hypersurface 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.

Titles in this series are co-published with International Press of Boston, Inc., Cambridge, MA.

#### Readership

Graduate students and research mathematicians interested in differential geometry, partial differential equations, and related applications; engineers, chemists, and biologists.

# Table of Contents

## Lectures on Mean Curvature Flows

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- The curve shortening flow for convex curves 112
- The short time existence and the evolution equation of curvatures 1526
- Contraction of convex hypersurfaces 2536
- Monotonicity and self-similar solutions 3546
- Evolution of embedded curves or surfaces (I) 4758
- Evolution of embedded curves and surfaces (II) 5566
- Evolution of embedded curves and surfaces (III) 6778
- Convexity estimates for mean convex surfaces 7788
- Li-Yau estimates and type II singularities 89100
- The mean curvature flow in Riemannian manifolds 101112
- Contracting convex hypersurfaces in Riemannian manifolds 109120
- Definition of center of mass for isolated gravitating systems 123134
- References 145156
- Index 149160
- Back Cover Back Cover1162