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

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) CAT (0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534 and obtaining same regularity as in H. -C. Zhang and X. -P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934 for certain sub-Riemannian geometries; see also N. Gigli, On the regularity of harmonic maps from RCD ⁢ (K, N) RCD (K, N) to CAT ⁢ (0) CAT (0) spaces and related results, preprint (2022), https: //arxiv. org/abs/2204. 04317 ; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ (k, n) RCD (k, n) spaces to CAT ⁡ (0) CAT (0) spaces, preprint (2022), https: //arxiv. org/abs/2202. 01590 for the generalisation 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.