Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) spaces

2023/12/28 by Renan Assimos, Assimos, Renan, Yaoting Gui +3
Mathematics · Physics and Astronomy · #31E05 #35B45 #35B65 #35D30 #51F99 #53C17 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2312.17112

openalex publication_date 2023/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven by Y. Gui et. al and obtaining same regularity as in the work of H-C. Zhang and X-P. Zhu for certain sub-Riemannian geometries, see also the works of A. Mondino and D. Semola, as well as N. Gigli, for generalisations to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

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