Harmonic maps and rigidity

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We examine harmonic maps into non-positively curved metric spaces, with a focus on their regularity when mapping into Euclidean buildings. We extend the regularity results of Gromov and Schoen by considering buildings that are not necessarily locally finite. As an application, we prove a superrigidity theorem for algebraic groups, which generalizes the rank 1 p-adic superrigidity results of Gromov and Schoen. This work also provides a geometric framework for Bader and Furman's generalization of Margulis' higher rank superrigidity theorem, bridging the gap between geometric and algebraic approaches to rigidity phenomena.

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