Regularity of harmonic maps to trees

Abstract

We derive regularity theorems and local finiteness results in this paper. We first prove the ∈-regularity type theorem for harmonic maps to R -trees. That is when the order of a harmonic map at a point is sufficiently close to 1 then this point is a regular point. We then prove a regularity theorem near higher order points. We show that if the image of a fixed ball of a properly normalized harmonic map is sufficiently close to a subtree, then the image of a smaller ball of this map lies in that subtree. As an application we prove that the local image of a harmonic map to an R -tree lies in a finite homogeneous subtree.