Hörmander systems and harmonic morphisms

Barletta, Elisabetta. "Hörmander systems and harmonic morphisms." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 379-394. <http://eudml.org/doc/84505>.

@article{Barletta2003,
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 (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 $\Omega = \{\bf H\}_n$ (the Heisenberg group) and $X = \left\lbrace \frac\{1\}\{2\}\left( L_\alpha + L_\{\overline\{\alpha \}\}\right) , \frac\{1\}\{2i\}\left( L_\alpha - L_\{\overline\{\alpha \}\}\right)\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 $\phi \circ \pi : (C(\{\bf H\}_n ) , F_\{\theta _0\} ) \rightarrow N$ is a harmonic morphism, where $S^1 \rightarrow C(\{\bf H\}_n ) \overset\{\pi \}\{\rightarrow \}\{\rightarrow \} \{\bf H\}_n$ is the canonical circle bundle and $F_\{\theta _0\}$ is the Fefferman metric of $(\{\bf H\}_n , \theta _0 )$. For any $S^1$-invariant weak solution to the harmonic map equation on $(C(\{\bf H\}_n ) , F_\{\theta _0\})$ the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from $(C(\lbrace x_1 &gt; 0 \rbrace ), F_\{\theta (k)\})$ into a riemannian manifold, where $F_\{\theta (k)\}$ 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 = \{\bf R\}^3 \setminus \lbrace x_1 = 0 \rbrace $.},
author = {Barletta, Elisabetta},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {379-394},
publisher = {Scuola normale superiore},
title = {Hörmander systems and harmonic morphisms},
url = {http://eudml.org/doc/84505},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Barletta, Elisabetta
TI - Hörmander systems and harmonic morphisms
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 379
EP - 394
AB - 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 (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 $\Omega = {\bf H}_n$ (the Heisenberg group) and $X = \left\lbrace \frac{1}{2}\left( L_\alpha + L_{\overline{\alpha }}\right) , \frac{1}{2i}\left( L_\alpha - L_{\overline{\alpha }}\right)\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 $\phi \circ \pi : (C({\bf H}_n ) , F_{\theta _0} ) \rightarrow N$ is a harmonic morphism, where $S^1 \rightarrow C({\bf H}_n ) \overset{\pi }{\rightarrow }{\rightarrow } {\bf H}_n$ is the canonical circle bundle and $F_{\theta _0}$ is the Fefferman metric of $({\bf H}_n , \theta _0 )$. For any $S^1$-invariant weak solution to the harmonic map equation on $(C({\bf H}_n ) , F_{\theta _0})$ the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from $(C(\lbrace x_1 &gt; 0 \rbrace ), F_{\theta (k)})$ into a riemannian manifold, where $F_{\theta (k)}$ 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 = {\bf R}^3 \setminus \lbrace x_1 = 0 \rbrace $.
LA - eng
UR - http://eudml.org/doc/84505
ER -