Luc Lemaire

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.

On the space of harmonic 2-spheres in CP2

Carrying further the work of T.A. Crawford, we show that each component of the space of harmonic maps from the 2-sphere to complex projective 2-space of degree d and energy 4πE is a smooth closed submanifold of the space of all Cj maps (j≥2). We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is smooth and has injective differential.