Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity

TL;DR

This paper extends regularity results for harmonic maps into Euclidean buildings and applies these findings to establish superrigidity for algebraic groups over non-Archimedean fields, generalizing previous results.

Contribution

It proves new regularity properties for harmonic maps into Euclidean buildings and applies these to establish superrigidity in a broader non-Archimedean context.

Findings

Harmonic maps into Euclidean buildings have singular sets of Hausdorff codimension 2.

Superrigidity for algebraic groups over non-Archimedean fields is established.

Existence of pluriharmonic maps from Kähler manifolds to Euclidean buildings is proved.

Abstract

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.