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<p>We prove a discreteness result for the possible orders of harmonic maps from surfaces to Euclidean buildings; in particular for a building of type <i>W</i> the order is of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{m}{k}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>m</mi> <mi>k</mi> </mfrac> </math></EquationSource> </InlineEquation> where <i>k</i> divides |<i>W</i>|. 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.</p>

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On the Possible Orders of Harmonic Maps into Euclidean Buildings

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • Christine Breiner,
  • Ben K. Dees

摘要

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 \(\frac{m}{k}\) m k 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.