On the Possible Orders of Harmonic Maps into Euclidean Buildings

Christine Breiner; Ben K. Dees

On the Possible Orders of Harmonic Maps into Euclidean Buildings

Differential Geometry 2026-04-21 v1

Authors: Christine Breiner , Ben K. Dees

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.

Keywords

harmonic maps Euclidean buildings discrete orders billiards problem geometric analysis

Cite

@article{arxiv.2604.16608,
  title  = {On the Possible Orders of Harmonic Maps into Euclidean Buildings},
  author = {Christine Breiner and Ben K. Dees},
  journal= {arXiv preprint arXiv:2604.16608},
  year   = {2026}
}

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12 pages

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