On the Possible Orders of Harmonic Maps into Euclidean Buildings

TL;DR

This paper establishes a discreteness result for the possible orders of harmonic maps from surfaces into Euclidean buildings, generalizing previous work by Gromov and Schoen.

Contribution

It proves that the order of such harmonic maps is of the form m/k where k divides the building's type order, extending known results to higher dimensions.

Findings

The order of harmonic maps is of the form m/k with k dividing |W|.

Analysis of homogeneous maps into Euclidean buildings is key.

A spherical billiards problem is used to study the behavior of these maps.

Abstract

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 $\frac mk$ where $k$ divides $|W|$. 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.