Lipschitz continuity of harmonic maps between Alexandrov spaces

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 Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder 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.

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

References in corpus (1)

• Pathological solutions to elliptic problems in divergence form with continuous coefficients

Cited by in corpus (1)

• On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications