Lipschitz Regularity of Harmonic Maps between Alexandrov Spaces

Alexandrov spaces are singular metric spaces with generalized lower or upper bounds of sectional curvature via Toponogov’s triangle comparison theorem. In 1992, to study the p-adic superrigidity for lattices in groups of rank one, M. Gromov and R. Schoen developed a theory of harmonic maps from smooth manifolds into nonpositively curved singular spaces.

In 1997, J. Jost and F. H. Lin proved that every harmonic map from an Alexandrov space with curvature bounded from below to a non-positive curvature (NPC) metric space is locally Hölder continuous. Based on Gromov-Schoen’s and Korevaar-Schoen’s results, F. H. Lin also conjectured that the Hölder continuity can be improved to Lipschitz continuity. J. Jost also asked a similar problem. In this talk, we will introduce a resolution to this Lin’s problem. This is a joint work with Prof. Xi-Ping Zhu.