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ON HARMONIC AND PSEUDOHARMONIC MAPS FROM PSEUDO-HERMITIAN MANIFOLDS

Part of: Global differential geometry CR manifolds

Published online by Cambridge University Press:  06 November 2017

TIAN CHONG ,
YUXIN DONG ,
YIBIN REN  and
GUILIN YANG
TIAN CHONG
Affiliation:
School of Science, College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, PR China email chongtian@sspu.edu.cn
YUXIN DONG
Affiliation:
School of Mathematical Science Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, PR China email yxdong@fudan.edu.cn
YIBIN REN
Affiliation:
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, PR China email allenry@outlook.com
GUILIN YANG
Affiliation:
School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, PR China email glyang@hust.edu.cn
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Abstract

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

MSC classification

Primary: 32V20: Analysis on CR manifolds 53C43: Differential geometric aspects of harmonic maps
Type
Article
Information
Nagoya Mathematical Journal , Volume 234 , June 2019 , pp. 170 - 210
Copyright
© 2017 Foundation Nagoya Mathematical Journal  

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References

Barletta, E., Dragomir, S. and Urakawa, H., Pseudoharmonic maps from nondegenrate CR manifolds to Riemannian manifolds , Indiana Univ. Math. J. 50(2) (2001), 719–746.10.1512/iumj.2001.50.1931Google Scholar
Boyer, C. P. and Galicki, K., 3-Sasakian manifolds, preprint, 1998, arXiv:hep-th/9810250.Google Scholar
Carlson, J. A. and Toledo, D., Harmonic mappings of Kähler manifolds to locally symmetric spaces , Publ. Math. Inst. Hautes Études Sci. 69(1) (1989), 173–201.10.1007/BF02698844Google Scholar
Chang, T.-H., The CR Bochner identity and stable pseudoharmonic maps on pseudohermitian manifolds, Thesis, National Tsing Hua University, Taiwan, 2010.Google Scholar
Chang, S.-C. and Chiu, H.-L., Nonnegativity of CR Paneitz operator and its application to the CR Obata’s theorem , J. Geom. Anal. 19 (2009), 261–287.10.1007/s12220-008-9060-9Google Scholar
Dong, Y. X., Monotonicity formula and holomorphicity of harmonic maps between Kähler manifolds , Proc. Lond. Math. Soc. (3) 107 (2013), 1221–1260.10.1112/plms/pdt014Google Scholar
Dong, Y. X., Pseudo-harmonic maps between Pseudo-Hermitian manifolds, arXiv:1610.01032.Google Scholar
Dragomir, S. and Kamishima, Y., Pseudoharmonic maps and vector fields on CR manifolds , J. Math. Soc. Japan 62 (2010), 269–303.10.2969/jmsj/06210269Google Scholar
Dragomir, S. and Tomassini, G., Differential Geometry and Analysis on CR Manifolds, Birkhauser, Boston, 2006.Google Scholar
Eells, J. and Lemaire, L., A report on harmonic maps , Bull. Lond. Math. Soc. 10 (1978), 1–68.10.1112/blms/10.1.1Google Scholar
Gherghe, C., Ianus, S. and Pastore, A. M., CR-manifolds, harmonic maps and stability , J. Geom. 71 (2001), 42–53.Google Scholar
Graham, C. R. and Lee, J. M., Smooth solutions of degenerate Laplaces on strictly pseudoconvex domains , Duke Math. J. 57 (1988), 697–720.10.1215/S0012-7094-88-05731-6Google Scholar
Ianus, S. and Pastore, A. M., Harmonic maps on contact metric manifolds , Ann. Math. Blaise Pascal 2(2) (1995), 43–53.10.5802/ambp.46Google Scholar
Lee, J. M., Pseudo-Einstein strucutre on CR manifolds , Amer. J. Math. 110 (1988), 157–178.10.2307/2374543Google Scholar
O’Neill, B., Semi-Riemannian Geometry: with Applications to Relativity, Pure and Applied Math. 103 , Academic Press, New York, 1983.Google Scholar
Petit, R., Harmonic maps and strictly pseudoconvex CR manifolds , Comm. Anal. Geom. 10(3) (2002), 575–610.10.4310/CAG.2002.v10.n3.a5Google Scholar
Pigola, S., Rigoli, M. and Setti, A. G., Vanishing and Finiteness Results in Geometric Analysis, Progress in Mathematics 266 , Birkhauser, Basel, 2008.Google Scholar
Rigoli, M. and Setti, A. G., Liouville-type theorems for 𝜑-subharmonic functions , Rev. Mat. Iberoam. 17 (2001), 471–520.Google Scholar
Sampson, J. H., Applications of harmonic maps to Kähler geometry , Contemp. Math. 49 (1986), 125–134.10.1090/conm/049/833809Google Scholar
Siu, Y.-T., The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds , Ann. of Math. (2) 112 (1980), 73–111.10.2307/1971321Google Scholar
Siu, Y.-T., Curvature charecterization of hyperquadrics , Duke Math. J. 47 (1980), 641–654.Google Scholar
Tanaka, N., A Differential Geometry Study on Strictly Pseudo-convex Manifolds, Kinokuniya Book-Store, Tokyo, 1975.Google Scholar
Toledo, D., “ Rigidity Theorems in Kähler Geometry and Fundamental Groups of Varieties ”, in Several Complex Variables 37 , MSRI Publications, 1999.Google Scholar
Udagawa, S., Holomorphicity of certain stable harmonic maps and minimal immersions , Proc. Lond. Math. Soc. (3) 57 (1988), 577–598.10.1112/plms/s3-57.3.577Google Scholar
Urakawa, H., “ Variational problems over strongly pseudoconvex CR manifolds ”, in Differential Geometry, Proceedings of the Symposium in Honour of Prof. Su Buchin on His 90th Birthday, Shanghai, China, 17–23 Sept., 1991 (eds. Gu, C. H., Hu, H. S. and Xin, Y. L.) World Scientific, Singapor, New Jersey, London, HongKong, 1993, 233–242.Google Scholar
Webster, S. M., Pseudo-Hermitian strucures on a real hypersurface , J. Differential Geom. 13 (1978), 25–41.10.4310/jdg/1214434345Google Scholar