**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2002; 209 pp; Softcover

MSC: Primary 53; 58;

**Print ISBN: 978-0-8218-1356-0**

Product Code: MMONO/205

Product Code: MMONO/205

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-4630-7
Product Code: MMONO/205.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

# Variational Problems in Geometry

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*Seiki Nishikawa*

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.

#### Readership

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

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Variational Problems in Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the English edition vii8
- Preface ix10
- Outlines and objectives of the theory xiii14
- Arc-length of curves and geodesics 120
- First and second variation formulas 4766
- Energy of maps and harmonic maps 85104
- Existence of harmonic maps 119138
- Appendix A. Fundamentals of the theory of manifolds and functional analysis 163182
- Prospects for contemporary mathematics 183202
- Solutions to exercise problems 187206
- Bibliography 201220
- Books 203222
- Index 207226
- Copying and reprinting notice 211230
- Back Cover Back Cover1234