Given a Hörmander system $X = \{ X_1 , \dots , X_m \}$ on a domain $\Omega \subseteq \mathbb{R}^n$ 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 $\nu \leq m$ and $X$ has rank $m$. If $\Omega = \mathbb{H}_n$ (the Heisenberg group) and $X = \{ \frac{1}{2} (L_\alpha + L_{\bar \alpha} ) , \frac{1}{2} ( L_\alpha - L_{\bar \alpha} ) \}$, where $L_{\bar \alpha} = \partial /\partial {\bar z}^\alpha - i z^\alpha \partial /\partial t$ is the Lewy operator, then a smooth map $\phi : \Omega to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi : ( C(\mathbb{H}_n , F_{\theta_0} ) \to N$ is a harmonic morphism, where $S^1 \to C( \mathbb{H}_n ) {\buildrel{\pi} \over {\to}} \mathbb{H}_n$ is the canonical circle bundle and $F_{\theta_0}$ is the Fefferman metric of $(\mathbb{H}_n , \theta_0)$. For any $S^1$-invariant weak solution to the harmonic map equation on $( C(\mathbb{H}_n) , F_{\theta_0} ) the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for a weak harmonic morphism from $C(\{x_1 >0\}), F_{\theta (k)})$ into a Riemannian manifold, where $F_{\theta 8k)}$ is the fefferman metric associated to the system of vector fields $X_1 = \partial /\partial x_1, X_2 = \partial/\partial x_2 + x_1^k \partial /\partial x_3$ ($k\geq 1$) on $\Omega = \mathbb{R}^3 \setminus \{ x_1=0\}$.

Hörmander systems and harmonic morphisms

BARLETTA, Elisabetta
2003-01-01

Abstract

Given a Hörmander system $X = \{ X_1 , \dots , X_m \}$ on a domain $\Omega \subseteq \mathbb{R}^n$ 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 $\nu \leq m$ and $X$ has rank $m$. If $\Omega = \mathbb{H}_n$ (the Heisenberg group) and $X = \{ \frac{1}{2} (L_\alpha + L_{\bar \alpha} ) , \frac{1}{2} ( L_\alpha - L_{\bar \alpha} ) \}$, where $L_{\bar \alpha} = \partial /\partial {\bar z}^\alpha - i z^\alpha \partial /\partial t$ is the Lewy operator, then a smooth map $\phi : \Omega to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi : ( C(\mathbb{H}_n , F_{\theta_0} ) \to N$ is a harmonic morphism, where $S^1 \to C( \mathbb{H}_n ) {\buildrel{\pi} \over {\to}} \mathbb{H}_n$ is the canonical circle bundle and $F_{\theta_0}$ is the Fefferman metric of $(\mathbb{H}_n , \theta_0)$. For any $S^1$-invariant weak solution to the harmonic map equation on $( C(\mathbb{H}_n) , F_{\theta_0} ) the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for a weak harmonic morphism from $C(\{x_1 >0\}), F_{\theta (k)})$ into a Riemannian manifold, where $F_{\theta 8k)}$ is the fefferman metric associated to the system of vector fields $X_1 = \partial /\partial x_1, X_2 = \partial/\partial x_2 + x_1^k \partial /\partial x_3$ ($k\geq 1$) on $\Omega = \mathbb{R}^3 \setminus \{ x_1=0\}$.
2003
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/3087
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