Hörmander systems and harmonic morphisms
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, p. 379-394

Given a Hörmander system X={X 1 ,,X m } on a domain Ω𝐑 n we show that any subelliptic harmonic morphism φ from Ω into a ν-dimensional riemannian manifold N is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also φ is a submersion provided that νm and X has rank m. If Ω=𝐇 n (the Heisenberg group) and X=1 2L α +L α ¯ ,1 2iL α -L α ¯ , where L α ¯ =/z ¯ α -iz α /t is the Lewy operator, then a smooth map φ:ΩN is a subelliptic harmonic morphism if and only if φπ:(C(𝐇 n ),F θ 0 )N is a harmonic morphism, where S 1 C(𝐇 n ) π𝐇 n is the canonical circle bundle and F θ 0 is the Fefferman metric of (𝐇 n ,θ 0 ). For any S 1 -invariant weak solution to the harmonic map equation on (C(𝐇 n ),F θ 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({x 1 >0}),F θ(k) ) into a riemannian manifold, where F θ(k) is the Fefferman metric associated to the system of vector fields X 1 =/x 1 ,X 2 =/x 2 +x 1 k /x 3 (k1) on Ω=𝐑 3 {x 1 =0}.

Classification:  58E20,  53C43,  32V20,  35H20
@article{ASNSP_2003_5_2_2_379_0,
     author = {Barletta, Elisabetta},
     title = {H\"ormander systems and harmonic morphisms},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {2},
     year = {2003},
     pages = {379-394},
     zbl = {1170.58305},
     mrnumber = {2005608},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_2_379_0}
}
Barletta, Elisabetta. Hörmander systems and harmonic morphisms. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 379-394. http://www.numdam.org/item/ASNSP_2003_5_2_2_379_0/

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