Hörmander systems and harmonic morphisms
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 379-394.

Given a Hörmander system $X=\left\{{X}_{1},\cdots ,{X}_{m}\right\}$ on a domain $\Omega \subseteq {𝐑}^{n}$ we show that any subelliptic harmonic morphism $\phi$ from $\Omega$ into a $\nu$-dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $\phi$ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega ={𝐇}_{n}$ (the Heisenberg group) and $X=\left\{\frac{1}{2}\left({L}_{\alpha }+{L}_{\overline{\alpha }}\right),\frac{1}{2i}\left({L}_{\alpha }-{L}_{\overline{\alpha }}\right)\right\}$, where ${L}_{\overline{\alpha }}=\partial /\partial {\overline{z}}^{\alpha }-i{z}^{\alpha }\partial /\partial t$ is the Lewy operator, then a smooth map $\phi :\Omega \to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :\left(C\left({𝐇}_{n}\right),{F}_{{\theta }_{0}}\right)\to N$ is a harmonic morphism, where ${S}^{1}\to C\left({𝐇}_{n}\right)\stackrel{\pi }{\to }\to {𝐇}_{n}$ is the canonical circle bundle and ${F}_{{\theta }_{0}}$ is the Fefferman metric of $\left({𝐇}_{n},{\theta }_{0}\right)$. For any ${S}^{1}$-invariant weak solution to the harmonic map equation on $\left(C\left({𝐇}_{n}\right),{F}_{{\theta }_{0}}\right)$ the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from $\left(C\left(\left\{{x}_{1}>0\right\}\right),{F}_{\theta \left(k\right)}\right)$ into a riemannian manifold, where ${F}_{\theta \left(k\right)}$ is the Fefferman metric associated to the system of vector fields ${X}_{1}=\partial /\partial {x}_{1},{X}_{2}=\partial /\partial {x}_{2}+{x}_{1}^{k}\phantom{\rule{0.277778em}{0ex}}\partial /\partial {x}_{3}$ $\phantom{\rule{0.277778em}{0ex}}\left(k\ge 1\right)$ on $\Omega ={𝐑}^{3}\setminus \left\{{x}_{1}=0\right\}$.

Classification: 58E20,  53C43,  32V20,  35H20
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title = {H\"ormander systems and harmonic morphisms},
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Barletta, Elisabetta. Hörmander systems and harmonic morphisms. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 379-394. http://www.numdam.org/item/ASNSP_2003_5_2_2_379_0/

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