Extrinsic geometric flows / Ben Andrews [and three others].

Format:

Book

Author/Creator:

Andrews, Ben, author.

Chow, Bennett, author.

Guenther, Christine (Christine Marie), 1966- author.

Langford, Mat (Mathew), 1987- author.

Series:

Graduate studies in mathematics ; Volume 206.

Graduate Studies in Mathematics ; Volume 206

Language:

English

Subjects (All):

Global differential geometry.

Differential equations, Parabolic.

Flows (Differentiable dynamical systems).

Curvature.

Geometric analysis.

Physical Description:

1 online resource (xxviii, 759 pages) : illustrations.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2020.

Language Note:

English

Summary:

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 inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean 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.

Contents:

The heat equation

Introduction to curve shortening

The Gage-Hamilton and Grayson theorems

Self-similar 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

Self-similar solutions

Ancient solutions

Gauss curvature flows

The affine normal flow

Flows by super-affine powers of the Gauss curvature

Fully nonlinear curvature flows

Flows of mean curvature type

Flows of inverse-mean curvature type.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781470456863

1470456869

OCLC:

1159163538