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.

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
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     title = {H\"ormander systems and harmonic morphisms},
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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/

[1] E. Barletta - S. Dragomir - H. Urakawa, Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds, Indiana Univ. Math. J. 50 (2) (2001), 719-746. | MR | Zbl

[2] E. Barletta - S. Dragomir, Differential equations on contact Riemannian manifolds, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 30 (2001), 63-95. | EuDML | Numdam | MR | Zbl

[3] A. Besse, “Einstein manifolds”, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo, 1987. | MR | Zbl

[4] S. Dragomir, A survey of pseudohermitian geometry, Proceedings of the Workshop on Differential Geometry and Topology (Palermo 1996), Rend. Circ. Mat. Palermo Suppl. 49 (2) (1997), 101-112. | MR | Zbl

[5] B. Fuglede, Harmonic morphisms between semi-Riemannian manifolds, Ann. Acad. Sci. Fenn. Math. 21 (2) (1996), 31-50. | EuDML | MR | Zbl

[6] S. Helgason, “Groups and geometric analysis”, Academic Press Inc., New York, London, Tokyo, 1984. | MR | Zbl

[7] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (2) (1979), 215-229. | MR | Zbl

[8] J. Jost, “Riemannian geometry and geometric analysis”, Springer, Berlin-Heidelberg, 2002 (third edition). | MR | Zbl

[9] J. Jost - C-J. Xu, Subelliptic harmonic maps, Trans. Amer. Math. Soc. 350 (11) (1998), 4633-4649. | MR | Zbl

[10] J.M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc. 296 (1) (1986), 411-429. | MR | Zbl

[11] N. Tanaka, “A differential geometric study on strongly pseudo-convex manifolds”, Kinokuniya Book Store Co., Ltd., Kyoto, 1975. | MR | Zbl

[12] S.M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geometry 13 (1978), 25-41. | MR | Zbl

[13] J.C. Wood, Harmonic morphisms, foliations and Gauss maps, Contemp. Math. 49 (1986), 145-184. | MR | Zbl

[14] C-J. Xu - C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var. Partial Differential Equations 5 (4) (1997), 323-343. | MR | Zbl

[15] Z-R. Zhou, Uniqueness of subelliptic harmonic maps, Ann. Global Anal. Geom. 17 (6) (1999), 581-594. | MR | Zbl