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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CAT</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>RCD</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{RCD}(K,N) to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CAT</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>RCD</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{RCD}(k,n) spaces to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CAT</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{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.