arXiv subjects

Ben K. Dees

Publications and source records attributed to Ben K. Dees.

3 recordsLinked to original sources

arXiv2026

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 $\frac mk$ 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.

math.DG↗

arXiv2025

Rectifiability of the singular strata for harmonic maps to Euclidean buildings

We define a natural notion of the singular strata for harmonic maps into $F$-connected complexes (which include locally finite Euclidean buildings), and prove the rectifiability of these strata. We additionally establish bounds on the Minkowski content for certain quantitative strata, following the rectifiable Reifenberg program of [NV17]. This builds on a result of the second author [D], which showed that the full singular set is $(n-2)$-rectifiable.

math.DG↗

arXiv2024

Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity

We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove superrigidity for algebraic groups over fields with non-Archimedean valuation, thereby generalizing the rank 1 $p$-adic superrigidity results of Gromov and Schoen and casting the Bader-Furman generalization of Margulis' higher rank superrigidity result in a geometric setting. We also prove an existence theorem for a pluriharmonic map from a K\"ahler manifold to a Euclidean building.

math.DG↗