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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.
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Curve Shortening Problem The © 2001 by Chapman & Hall/CRC Curve Shortening Problem The Kai-Seng Chou Xi-Ping Zhu CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C. © 2001 by Chapman & Hall/CRC Library of Congress Cataloging-in-Publication Data Chou, Kai Seng. The curve shortening problem / Kai-Seng Chou, Xi-Ping Zhu. p. cm. Includes bibliographical references and index. ISBN 1-58488-213-1 (alk. paper) 1. Curves on surfaces. 2. Flows (Differentiable dynamical systems) 3. Hamiltonian sytems. I. Zhu, Xi-Ping. II. Title. QA643 .C48 2000 516.3′52—dc21 00-048547 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have bee