Hörmander systems and harmonic morphisms

Barletta, Elisabetta

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 379-394.

Given a Hörmander system $X=\{X_1,\cdots ,X_m\}$ 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 $\phi \circ \pi :(C\left({𝐇}_n\right),F_{{\theta }_0})\to N$ 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 $({𝐇}_n,{\theta }_0)$. For any $S^1$-invariant weak solution to the harmonic map equation on $(C\left({𝐇}_n\right),F_{{\theta }_0})$ the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from $(C\left(\{x_1>0\}\right),F_{\theta \left(k\right)})$ into a riemannian manifold, where $F_{\theta \left(k\right)}$ is the Fefferman metric associated to the system of vector fields $X_1=\partial /\partial x_1,X_2=\partial /\partial x_2+x_1^k\partial /\partial x_3$ $(k\ge 1)$ on $\Omega ={𝐑}^3\setminus \{x_1=0\}$.

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