Brakke's Mean Curvature Flow An Introduction
This book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimens
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Yoshihiro Tonegawa
Brakke’s Mean Curvature Flow An Introduction
123
SpringerBriefs in Mathematics Series editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel George Yin Ping Zhang Luis Gustavo Nonato Paulo J. S. Silva
SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians.
More information about this series at http://www.springer.com/series/10030
Yoshihiro Tonegawa
Brakke’s Mean Curvature Flow An Introduction
123
Yoshihiro Tonegawa Tokyo Institute of Technology Tokyo, Japan
ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-981-13-7074-8 ISBN 978-981-13-7075-5 (eBook) https://doi.org/10.1007/978-981-13-7075-5 Library of Congress Control Number: 2019934770 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
To Lili, Sana, Suzue, and Yoshitsugu.
Preface
The aim of this short book is to introduce to graduate students and researchers a notion of mean curvature flow created by Ken Brakke in [7], which is the expanded book version published in 1978 of his Ph.D. thesis under his supervisor Fred Almgren. At present, this notion of mean curvature flow, which is often called the Brakke flow, is little known outside of a small circle of specia
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