The Curve Shortening Problem: 1st Edition (Hardback) book cover

The Curve Shortening Problem

1st Edition

By Kai-Seng Chou, Xi-Ping Zhu

Chapman and Hall/CRC

272 pages | 10 B/W Illus.

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Hardback: 9781584882138
pub: 2001-03-06
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pub: 2001-03-06
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Description

Although research in curve shortening flow has been very active for nearly 20 years, the results of those efforts have remained scattered throughout the literature. For the first time, The Curve Shortening Problem collects and illuminates those results in a comprehensive, rigorous, and self-contained account of the fundamental results.

The authors present a complete treatment of the Gage-Hamilton theorem, a clear, detailed exposition of Grayson's convexity theorem, a systematic discussion of invariant solutions, applications to the existence of simple closed geodesics on a surface, and a new, almost convexity theorem for the generalized curve shortening problem.

Many questions regarding curve shortening remain outstanding. With its careful exposition and complete guide to the literature, The Curve Shortening Problem provides not only an outstanding starting point for graduate students and new investigations, but a superb reference that presents intriguing new results for those already active in the field.

Table of Contents

BASIC RESULTS

Short Time Existence

Facts from Parabolic Theory

Evolution of Geometric Quantities

INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW

Travelling Waves

Spirals

The Support Function of a Convex Curve

Self-Similar Solutions

THE CURVATURE-EIKONAL FLOW FOR CONVEX CURVES

Blaschke Selection Theorem

Preserving Convexity and Shrinking to a Point

Gage-Hamilton Theorem

The Contracting Case of the ACEF

The Stationary case of the ACEF

The Expanding Case of the ACEF

THE CONVEX GENERALIZED CURVE SHORTENING FLOW

Results from Brunn-Minkowski Theory

The AGCSF for s in (1/3,1)

The Affine Curve Shortening Flow

Uniqueness of Self-Similar Solutions

THE NON-CONVEX CURVE SHORTENING FLOW

An Isoperimetric Ratio

Limits of the Rescaled Flow

Classification of Singularities

A CLASS OF NON-CONVEX ANISOTROPIC FLOWS

Decrease in Total Absolute Curvature

Existence of a Limit Curve

Shrinking to a Point

A Whisker Lemma

The Convexity Theorem

EMBEDDED CLOSED GEODESICS ON SURFACES

Basic Results

The Limit Curve

Shrinking to a Point

Convergence to a Geodesic

THE NON-CONVEX GENERALIZED CURVE SHORTENING FLOW

Short Time Existence

The Number of Convex Arcs

The Limit Curve

Removal of Interior Singularities

The Almost Convexity Theorem

BIBLIOGRAPHY

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT012000
MATHEMATICS / Geometry / General