Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence

This paper uses probability theory to provide an approach to the Dirichlet problem for harmonic maps. The probabilistic tools used are those of manifold-valued Brownian motion and Γ-martingales. Probabilistic proofs are given of uniqueness, continuous dependence on data, and existence of generalized solutions (finely harmonic maps), for target domains which have convex geometry. (This class of domain includes all regular geodesic balls.) Links are made with new results in other fields: the Γ-martingale Dirichlet problem, Riemannian centres of mass (herein termed ‘Karcher means’), and the existence of certain convex functions. Future prospects include the development of the probabilistic approach in order to formulate the notion of a harmonic map defined on a fractal and to attack the corresponding Dirichlet problem.