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Variational Problems in Geometry

Seiki Nishikawa Mathematical Institute, Tohoku University, Sendai, Japan
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Softcover ISBN: 978-0-8218-1356-0
Product Code: MMONO/205
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AMS Member Price: $44.00 Electronic ISBN: 978-1-4704-4630-7 Product Code: MMONO/205.E List Price:$55.00
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AMS Member Price: $66.00 Click above image for expanded view Variational Problems in Geometry Seiki Nishikawa Mathematical Institute, Tohoku University, Sendai, Japan Available Formats:  Softcover ISBN: 978-0-8218-1356-0 Product Code: MMONO/205  List Price:$55.00 MAA Member Price: $49.50 AMS Member Price:$44.00
 Electronic ISBN: 978-1-4704-4630-7 Product Code: MMONO/205.E
 List Price: $55.00 MAA Member Price:$49.50 AMS Member Price: $44.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$82.50 MAA Member Price: $74.25 AMS Member Price:$66.00
• Book Details

Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics
Volume: 2052002; 209 pp
MSC: Primary 53; 58;

A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus.

Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge.

This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps.

The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an in-depth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation.

The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of Eells-Sampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as self-contained as possible for easy reading.

Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter.

This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.

Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, global analysis, and analysis on manifolds.

• Chapters
• Arc-length of curves and geodesics
• First and second variation formulas
• Energy of maps and harmonic maps
• Existence of harmonic maps
• Appendix A. Fundamentals of the theory of manifolds and functional analysis
• Prospects for contemporary mathematics
• Reviews

• A welcome contribution to and survey of some of the fundamental questions and results in geometric variational problems on the length of curves and the energy of maps.

Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Iwanami Series in Modern Mathematics
Volume: 2052002; 209 pp
MSC: Primary 53; 58;

A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus.

Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge.

This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps.

The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an in-depth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation.

The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of Eells-Sampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as self-contained as possible for easy reading.

Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter.

This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.

Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, global analysis, and analysis on manifolds.

• Chapters
• Arc-length of curves and geodesics
• First and second variation formulas
• Energy of maps and harmonic maps
• Existence of harmonic maps
• Appendix A. Fundamentals of the theory of manifolds and functional analysis
• Prospects for contemporary mathematics
• A welcome contribution to and survey of some of the fundamental questions and results in geometric variational problems on the length of curves and the energy of maps.

Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.