James Eells

A setting for global analysis

Introduction. The primary aim of this report is to present a broad outline for a coherent geometric theory of certain aspects of nonlinear functional analysis. Its setting requires the calculus in topological vector spaces, differential geometry of infinite dimensional manifolds, and the algebraic and differential topology of function spaces. For the most part the developments are of quite recent origin, and at present the theory is in a fluid state (its growth depending strongly on its concrete applications). The beginnings of the subject may be traced to the work of Fréchet, Gâteaux, and Vol terra; we refer to the text [73] of P. Levy for an exposition of some early applications (especially in the calculus of variations and integrable differential systems)—and ask pardon for not presenting any historical perspective in the present survey. About ten years ago it was formally recognized [29] that many of the function spaces which arise in global geometric mathematics possess a natural infinite dimensional differentiable manifold structure. Tha t was not a great surprise; for (1) Many of the most interesting manifolds of differential geometry are well known to have representations as function spaces of rigid maps. (E.g., Riemannian manifolds arise as the configuration spaces of dynamical systems, their cotangent bundles are interpreted as phase spaces, and their Riemannian metrics in terms of kinetic energy.) (2) Much of the language of the classical treatment of the calculus of variations—and the penetrating viewpoint and methods of M. Morse—is that of a function space differential geometry. (E.g., the Euler-Lagrange operator of a variational problem has an interpretation as a gradient vector field, whose trajectories are lines of steepest descent.) (3) Certain eigenvalue problems in integral and differential equations have interpretations in terms of Lagranges method of multipliers, involving differential geometric ideas in infinite dimensions (e.g., focal point theory, and geometric consequences of the inverse

Harmonic maps from surfaces to complex projective spaces

In [ 171 the authors showed that any harmonic map from a compact Riemann surface MP of genus p to the sphere M, is holomorphic, provided its degree is greater than or equal to p. This gave the first indications of a possible classification theorem for certain harmonic maps in terms of holomorphic maps; it also showed that there exists no harmonic map of degree one from the torus M, to the sphere M,, whatever Riemannian metrics they may be given. It was clear from the outset that our methods (especially, an analysis of holomorphic quadratic differentials on MP) required that the domain be two dimensional. On the other hand, it was less obvious whether our theorem required the range to be two dimensional. In the present paper we shall replace the sphere M, = CP’ by the complex projective n-space CIP” with its Fubini-study metric; and show in particular if n > 2:

The surfaces of Delaunay

In 1841 the astronomer/mathematician C. Delaunay isolated a certain class of surfaces in Euclidean space, representations of which he described explicitly [1]. In an appendix to that paper, M. Sturm characterized Delaunay’s surfaces variationally; indeed, as the solutions to an isoperimetric problem in the calculus of variations. That in turn revealed how those surfaces make their appearance in gas dynamics; soap bubbles and stems of plants provide simple examples. See Chapter V of the marvellous book [8] by D’Arcy Thompson for an essay on the occurrence and properties of such surfaces in nature.

Two reports on harmonic maps

Harmonic maps between Riemannian manifolds are solutions of systems of nonlinear partial differential equations which appear in different contexts of differential geometry. They include holomorphic maps, minimal surfaces, -models in physics. Recently, they have become powerful tools in the study of global properties of Riemannian and Kahlerian manifolds.A standard reference for this subject is a pair of Reports, published in 1978 and 1988 by James Eells and Luc Lemaire.This book presents these two reports in a single volume with a brief supplement reporting on some recent developments in the theory. It is both an introduction to the subject and a unique source of references, providing an organized exposition of results spread throughout more than 800 papers.

Variational Theory in Fibre Bundles: Examples

Nature is uncompromisingly efficient. That article of faith has shaped much of our physical and geometric thinking; and any mathematical or physical model of Nature must display prominently its essential features. Indeed, systematic development of such guidelines into sound mathematical theory has -in large measure-led to one of our most powerful tools: The variational principle.