Uniqueness of the Boundary Value Problem of Harmonic Maps via Harmonic Boundary

Abstract

We prove the uniqueness of solutions for the boundary value problem of harmonic maps in the setting: given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a regular geodesic ball, there exists a unique harmonic map, which is a limit of a sequence of harmonic maps with finite total energy in the sense of the supremum norm, from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.