Rectifiability of the Singular Set of Harmonic Maps into Buildings

Abstract

In Gromov and Schoen’s (Publications Mathématiques de l’Institut des Hautes Études Scientifiques 76(1):165–246, 1992), a notion of energy-minimizing maps into singular spaces is developed. Using some analogs of classical tools, such as Almgren’s frequency function, they focus on harmonic maps into F-connected complexes, and demonstrate that the singular sets of such functions are closed and of codimension 2. More recent work on related problems, such as Naber and Valtorta’s (Ann Math 185(1):131–227, 2017), uses the study of monotone quantities such as the frequency function to prove stronger rectifiability results about the singular sets in these contexts. In this paper, we adapt this more recent work to demonstrate that an energy-minimizing map from a Euclidean space ${\mathbb {R}}^m$ to an F-connected complex has a singular set which is $(m-2)$-rectifiable. We also obtain Minkowski bounds on the singular sets of these maps, and in particular, use these bounds to demonstrate the local finiteness of the $(m-2)$-dimensional Hausdorff measure of these singular sets.