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Lectures on Mean Curvature Flows

Xi-Ping Zhu Zhongshan University, Guangzhou, People’s Republic of China
A co-publication of the AMS and International Press of Boston
Available Formats:
Hardcover ISBN: 978-0-8218-3311-7
Product Code: AMSIP/32
List Price: $51.00 MAA Member Price:$45.90
AMS Member Price: $40.80 Electronic ISBN: 978-1-4704-3821-0 Product Code: AMSIP/32.E List Price:$48.00
MAA Member Price: $43.20 AMS Member Price:$38.40
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $76.50 MAA Member Price:$68.85
AMS Member Price: $61.20 Click above image for expanded view Lectures on Mean Curvature Flows Xi-Ping Zhu Zhongshan University, Guangzhou, People’s Republic of China A co-publication of the AMS and International Press of Boston Available Formats:  Hardcover ISBN: 978-0-8218-3311-7 Product Code: AMSIP/32  List Price:$51.00 MAA Member Price: $45.90 AMS Member Price:$40.80
 Electronic ISBN: 978-1-4704-3821-0 Product Code: AMSIP/32.E
 List Price: $48.00 MAA Member Price:$43.20 AMS Member Price: $38.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$76.50 MAA Member Price: $68.85 AMS Member Price:$61.20
• Book Details

AMS/IP Studies in Advanced Mathematics
Volume: 322002; 150 pp
MSC: Primary 53; Secondary 35; 52; 58;

“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.

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

• Chapters
• 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
• Request Review Copy
Volume: 322002; 150 pp
MSC: Primary 53; Secondary 35; 52; 58;

“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.

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

• Chapters
• 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
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