Hörmander systems and harmonic morphisms

• Elisabetta Barletta ^[1]
  1. [1] Università della Basilicata

     Università della Basilicata

     Potenza, Italia

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 2, Nº 2, 2003, págs. 379-394
• Idioma: inglés
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• Resumen

  • 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 $S^1-invariant weak solution to the harmonic map equation on (C({\bf H}_n ) , F_{\theta _0}) the corresponding base map is shown to be a weak subelliptic harmonic map. We obtain a regularity result for weak harmonic morphisms from (C(\lbrace x_1 > 0 \rbrace ), F_{\theta (k)}) into a riemannian manifold, where F_{\theta (k)} is the Fefferman metric associated to the system of vector fields X_1 =\partial /\partial x_1 , X_2 = \partial /\partial x_2 + x_1^k \; \partial /\partial x_3$ \; (k \ge 1) on \Omega = {\bf R}^3 \setminus \lbrace x_1 = 0 \rbrace .