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Softcover ISBN:  9781470464578 
Product Code:  GSM/206.S 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $78.40 
Sale Price:  $63.70 
eBook ISBN:  9781470456863 
Product Code:  GSM/206.E 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $78.40 
Sale Price:  $63.70 
Softcover ISBN:  9781470464578 
eBook ISBN:  9781470456863 
Product Code:  GSM/206.S.B 
List Price:  $196.00 $147.00 
MAA Member Price:  $176.40 $132.30 
AMS Member Price:  $156.80 $117.60 
Sale Price:  $127.40 $95.55 

Book DetailsGraduate Studies in MathematicsVolume: 206; 2020; 790 ppMSC: Primary 53; 58; 52; 35
Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inversemean curvature flow, and fully nonlinear flows of mean curvature and inversemean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows.
The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.
ReadershipGraduate students and researchers interested in mean curvature flow.

Table of Contents

Chapters

The heat equation

Introduction to curve shortening

The Gage–Hamilton–Grayson theorem

Selfsimilar and ancient solutions

Hypersurfaces in Euclidean space

Introduction to mean curvature flow

Mean curvature flow of entire graphs

Huisken’s theorem

Mean convex mean curvature flow

Monotonicity formulae

Singularity analysis

Noncollapsing

Selfsimilar solutions

Ancient solutions

Gauß curvature flows

The affine normal flow

Flows by superaffine powers of the Gauß curvature

Fully nonlinear curvature flows

Flows of mean curvature type

Flows of inversemean curvature type


Additional Material

Reviews

This textbook, written by four experts in the field, offers an authoritative introduction and overview to the topic of extrinsic geometric flows. It will serve well as a primary text for a graduate student who already has background knowledge of differential geometry and (some) partial differential equations. It will also serve as a useful reference for experts in the field.
John Ross, Southwestern University


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Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inversemean curvature flow, and fully nonlinear flows of mean curvature and inversemean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows.
The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.
Graduate students and researchers interested in mean curvature flow.

Chapters

The heat equation

Introduction to curve shortening

The Gage–Hamilton–Grayson theorem

Selfsimilar and ancient solutions

Hypersurfaces in Euclidean space

Introduction to mean curvature flow

Mean curvature flow of entire graphs

Huisken’s theorem

Mean convex mean curvature flow

Monotonicity formulae

Singularity analysis

Noncollapsing

Selfsimilar solutions

Ancient solutions

Gauß curvature flows

The affine normal flow

Flows by superaffine powers of the Gauß curvature

Fully nonlinear curvature flows

Flows of mean curvature type

Flows of inversemean curvature type

This textbook, written by four experts in the field, offers an authoritative introduction and overview to the topic of extrinsic geometric flows. It will serve well as a primary text for a graduate student who already has background knowledge of differential geometry and (some) partial differential equations. It will also serve as a useful reference for experts in the field.
John Ross, Southwestern University