Minimality and Harmonicity in Differential Geometry

The theory of minimal surfaces and more generally of minimal immersions is one of the most fashionable branches in Diierential Geometry. Since the very early stage of the theory it was clear that there was a strong link between minimality and harmonicity. From a relation of E. Beltrami 4] follows that a surface in R 3 is minimal if and only if the components of the position vector are harmonic functions. In the context of minimal immersions, Eells{Sampson 10] proved that an isometric immersion is minimal if and only if it is a harmonic map. Harmonic morphisms form a subclass of harmonic maps. They are submersions and if the dimension of the codomain is two the inverse image of a point ((bre) is a minimal submanifold of the domain 2]. The aim of this lecture is to give a non-technical description of those features. Convention: We shall place ourselves in the C 1 category, i.e. all objects (mani-folds, metrics, connections, maps) are assumed C 1. 1. Minimal Surfaces The history of minimal surfaces begins with J.L. Lagrange. In his famous memoir 21] he developed the basic ideas of the calculus of variations. The examples discussed by Lagrange included the following.