Abstract

Given a Hörmander system $X = \lbrace X_1, \cdots, X_m \rbrace $ on a domain $\Omega \subseteq {\bf R}^n$ we show that any subelliptic harmonic morphism $\phi $ from $\Omega $ into a $\nu $-dimensional riemannian manifold $N$ is a subelliptic harmonic map. Also $\phi $ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega = {\bf H}_n$ and $X = \left\lbrace \frac{1}{2}\left, \frac{1}{2i}\left\right\rbrace $, where $L_{\overline{\alpha }} = \partial /\partial \overline{z}^\alpha - i z^\alpha \partial /\partial t$ is the Lewy operator, then a smooth map $\phi : \Omega \rightarrow N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :, F_{\theta _0} ) \rightarrow N$ is a harmonic morphism, where $S^1 \rightarrow C \overset{\pi }{\rightarrow }{\rightarrow } {\bf H}_n$ is the canonical circle bundle and $F_{\theta _0}$ is the Fefferman metric of $$. For any $S^1$-invariant weak solution to the harmonic map equation on $, F_{\theta _0})$ the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from $, F_{\theta })$ into a riemannian manifold, where $F_{\theta }$ is the Fefferman metric associated to the system of vector fields $X_1 =\partial /\partial x_1, X_2 = \partial /\partial x_2 + x_1^k \; \partial /\partial x_3$ $\; $ on $\Omega = {\bf R}^3 \setminus \lbrace x_1 = 0 \rbrace $