欧几里得建筑上的调和映射与非阿基米德超刚性
Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity
AI总结：
研究非阿基米德估值域上代数群的超刚性，扩展了Gromov和Schoen的局部有限正则性结果，并证明了Kähler流形到欧几里得建筑的多调和映射的存在性。
摘要：
我们证明了非局部有限的欧几里得建筑上的调和映射的奇集具有豪斯多夫余维数2，扩展了Gromov和Schoen的局部有限正则性结果。作为应用，我们证明了非阿基米德估值域上代数群的超刚性，从而推广了Gromov和Schoen的rank 1 $p$-adic超刚性结果，并将Bader-Furman对Margulis更高秩超刚性结果的推广置于几何背景下。我们还证明了从Kähler流形到欧几里得建筑的多调和映射的存在性定理。
英文摘要：
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 Kähler manifold to a Euclidean building.
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