On

Abstract

This paper considers maps from pseudo-Hermitian manifolds to Kähler manifolds and introduces partial energy functionals for these maps. First, the authors obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds by Shen–Shen–Zhang (2013). Next, the authors investigate critical maps of the partial energy functionals, which are referred to as

{\overline{\mathrm{\partial }}}_b

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline \partial}_{b}$$\end{document}

-harmonic maps and ∂_b-harmonic maps. The authors give a foliated result for both

{\overline{\mathrm{\partial }}}_b

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline \partial}_{b}$$\end{document}

- and ∂_b-harmonic maps, generalizing a foliated result of Petit (2002) for harmonic maps. Then the authors are able to generalize Siu’s holomorphicity result for harmonic maps by Siu (1980) to the case for

{\overline{\mathrm{\partial }}}_b

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline \partial}_{b}$$\end{document}

- and ∂_b-harmonic maps.

Keywords

Pseudo-Hermitian manifold /

{\overline{\mathrm{\partial }}}_b

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline \partial}_{b}$$\end{document}

-Harmonic maps')">

{\overline{\mathrm{\partial }}}_b

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline \partial}_{b}$$\end{document}

-Harmonic maps / Foliated CR map / 53C25 / 58E20

References

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

[2] Chong T, Dong Y X, Ren Y B, Yang G L. On harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds. Nagoya Math. J., 2019, 234: 170-210

[3] Dong, Y. X., On (H,H~)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(H, {\tilde H})$$\end{document}-harmonic maps between pseudo-Hermitian manifolds, 2016, arXiv:1610.01032.

[4] Dragomir S, Kamishima Y. Pseudoharmonic maps and vector fields on CR manifolds. J. Math. Soc. Japan, 2010, 62(1): 269-303

[5] Dragomir S, Tomassini G. Differential Geometry and Analysis on CR Manifolds, 2006, Boston, MA, Birkhäuser Boston, Inc.246

[6] Eells J, Lemaire L. Selected Topics in Harmonic Maps, 1983, Providence, RI, American Mathematical Society50

[7] Eells JJr., Sampson J H. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 1964, 86: 109-160

[8] Gherghe C, Ianus S, Pastore A M. CR-manifolds, harmonic maps and stability. J. Geom., 2001, 71(1–2): 42-53

[9] Graham C R, Lee J M. Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains. Duke Math. J., 1988, 57(3): 679-720

[10] Jost J. Nonlinear Methods in Riemannian and Kählerian Geometry, 1991second editionBasel, Birkhäuser Verlag 10

[11] Jost J, Yau S-T. A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math., 1993, 170(2): 221-254

[12] Li S-Y, Son D N. CR-analogue of the Siu-∂∂¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial{\overline \partial}$$\end{document}-formula and applications to the rigidity problem for pseudo-Hermitian harmonic maps. Proc. Amer. Math. Soc., 2019, 147(12): 5141-5151

[13] Lichnerowicz A. Applications harmoniques et variétés kähleriennes, 1970, London, New York, Academic Press341402III

[14] Liu K F, Yang X K. Hermitian harmonic maps and non-degenerate curvatures. Math. Res. Lett., 2014, 21(4): 831-862

[15] Petit R. Harmonic maps and strictly pseudoconvex CR manifolds. Comm. Anal. Geom., 2002, 10(3): 575-610

[16] Petit R. Mok-Siu-Yeung type formulas on contact locally sub-symmetric spaces. Ann. Global Anal. Geom., 2009, 35(1): 1-37

[17] Sampson J H. Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. (4), 1978, 11(2): 211-228

[18] Shen B, Shen Y B, Zhang X. Holomorphic maps from Sasakian manifolds into Kähler manifolds. Chin. Ann. Math. Ser. B, 2013, 34(4): 575-586

[19] Siu Y T. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math. (2), 1980, 112(1): 73-111

[20] Tanaka N. A differential geometric study on strongly pseudo-convex manifolds, 1975, Tokyo, Department of Mathematics, Kyoto University, Kinokuniya Book Store Co., Ltd.9

[21] Webster S M. Pseudo-Hermitian structures on a real hypersurface. J. Differential Geometry, 1978, 13(1): 25-41

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg