In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Diisseldorf, June, 1986. The title "Nonlinear methods in complex geometry" already indicates a combination of techniques from nonlinear partial differential equations and geometric concepts. In older geometric investigations, usually the local aspects attracted more attention than the global ones as differential geometry in its foundations provides approximations of local phenomena through infinitesimal or differential constructions. Here, all equations are linear. If one wants to consider global aspects, however, usually the presence of curvature leads to a nonlinearity in the equations. The simplest case is the one of geodesics which are described by a system of second order nonlinear ODE; their linearizations are the Jacobi fields. More recently, nonlinear PDE played a more and more prominent role in geometry. Let us list some of the most important ones: - harmonic maps between Riemannian and Kahlerian manifolds - minimal surfaces in Riemannian manifolds - Monge-Ampere equations on Kahler manifolds - Yang-Mills equations in vector bundles over manifolds. While the solution of these equations usually is nontrivial, it can lead to very signifi cant results in geometry, as solutions provide maps, submanifolds, metrics, or connections which are distinguished by geometric properties in a given context. All these equations are elliptic, but often parabolic equations are used as an auxiliary tool to solve the elliptic ones.

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.

This book is an exposition of what is currently known about the fundamental groups of compact Kähler manifolds. This class of groups contains all finite groups and is strictly smaller than the class of all finitely presentable groups. For the first time ever, this book collects together all the results obtained in the last few years which aim to characterize those infinite groups which can arise as fundamental groups of compact Kähler manifolds. Most of these results are negative ones, saying which groups don not arise. The methods and techniques used form an attractive mix of topology, differential and algebraic geometry, and complex analysis. The book would be useful to researchers and graduate students interested in any of these areas, and it could be used as a textbook for an advanced graduate course. One of its outstanding features is a large number of concrete examples. The book contains a number of new results and examples which have not appeared elsewhere, as well as discussions of some important open questions in the field.

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis

This text provides an introduction to the theory of partial differential equations. It introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, including particularly Fourier analysis, distribution theory, and Sobolev spaces. These tools are applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. Companion texts, which take the theory of partial differential equations further, are AMS volume 116, treating more advanced topics in linear PDE, and AMS volume 117, treating problems in nonlinear PDE. This book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.