Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

Abstract.

This is an addendum to the recent Cambridge Tract “Harmonic maps between Riemannian polyhedra”, by J. Eells and the present author. Hölder continuity of locally energy minimizing maps $\phi$ from an admissible Riemannian polyhedron X to a complete geodesic space Y is established here in two cases: (1) Y is simply connected and has curvature $\leq 0$ (in the sense of A.D. Alexandrov), or (2) Y is locally compact and has curvature $\leq1$, say, and $\phi(X)$ is contained in a convex ball in Y satisfying bi-point uniqueness and of radius $R<\pi/2$ (best possible). With Y a Riemannian polyhedron, and $R<\pi/4$ in case (2), this was established in the book mentioned above, though with Hölder continuity taken in a weaker, pointwise sense. For X a Riemannian manifold the stated results are due to N.J. Korevaar and R.M. Schoen, resp. T. Serbinowski.