Lipschitz continuity of harmonic maps between Alexandrov spaces
Differential Geometry
2017-09-08 v4 Analysis of PDEs
Metric Geometry
Abstract
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the H\"older continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
Keywords
Cite
@article{arxiv.1311.1331,
title = {Lipschitz continuity of harmonic maps between Alexandrov spaces},
author = {Hui-Chun Zhang and Xi-Ping Zhu},
journal= {arXiv preprint arXiv:1311.1331},
year = {2017}
}
Comments
We remove the assumption in the previous version that the domain space has nonnegative generalized Ricci curvature. This solves Lin's conjecture completely. To appear in Invent. Math