Harmonic maps into Euclidean buildings and non-Archimedean superrigidity

We prove that harmonic maps into Euclidean buildings, which include R-buildings, 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 Kähler manifold to a Euclidean building.1. Introduction 2251 2. Preliminaries 2256 3. Projection into the subbuilding defined by a flat 2264 4. Closeness in measure 2274 5. Homogeneous approximations 2280 6. Local product structure 2282 7. Proof of the main theorems 2287 Appendix Proof of Theorem 6.6 2289