On the Possible Orders of Harmonic Maps into Euclidean Buildings

We prove a discreteness result for the possible orders of harmonic maps from surfaces to Euclidean buildings; in particular for a building of type $W$ the order is of the form $\frac{m}{k}$ where $k$ divides $\mathrm{\mid }W\mathrm{\mid }$. This generalizes, in the case where the domain has dimension $2$, the “order gap” of Gromov and Schoen. This result follows by directly analyzing the behavior of homogeneous maps into Euclidean buildings, and then studying a related spherical billiards problem.