Chou / Zhu The Curve Shortening Problem

Erscheinungsjahr 2010
ISBN: 978-1-4200-3570-4
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 272 Seiten

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.

Zielgruppe

Graduate students and researchers in differential geometry, global analysis, and nonlinear evolution equations, graduate students and researchers in analysis and applied mathematicians

Autoren/Hrsg.

Chou, Kai-Seng

Zhu, Xi-Ping

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik

Inhaltsverzeichnis

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