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CR-HARMONIC MAPS

Published online by Cambridge University Press:  02 December 2019

GAUTIER DIETRICH*
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CNRS, Case courrier 051, Place Eugène Bataillon, 34090Montpellier, FranceUniversité Paul-Valéry Montpellier 3 email gautier.dietrich@ac-montpellier.fr

Abstract

We develop the notion of renormalized energy in Cauchy–Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.

Type
Article
Copyright
© 2019 Foundation Nagoya Mathematical Journal

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Footnotes

The author was supported in part by the grant ANR-17-CE40-0034 of the French National Research Agency ANR (project CCEM).

References

Bérard, V., Les applications conforme-harmoniques, Canad. J. Math. 65 (2013), 266298.CrossRefGoogle Scholar
Biquard, O., Métriques d’Einstein asymptotiquement symétriques, Astérisque 265, Société Mathématique de France, Paris, 2000.Google Scholar
Biquard, O. and Herzlich, M., A Burns–Epstein invariant for ACHE 4-manifolds, Duke Math. J. 126 (2005), 53100.CrossRefGoogle Scholar
Case, J. S. and Yang, P., A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.) 8(3) (2013), 285322.Google Scholar
Cheng, S.-Y. and Yau, S.-T., On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation, Commun. Pure Appl. Math. 33 (1980), 507544.CrossRefGoogle Scholar
Chong, T., Dong, Y., Ren, Y. and Yang, G., On harmonic and pseudoharmonic maps from pseudo-hermitian manifolds, Nagoya Math. J. 234 (2019), 170210.CrossRefGoogle Scholar
Dragomir, S. and Tomassini, G., Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics 246, Birkhäuser, Boston, 2006.Google Scholar
Epstein, C. L., Melrose, R. B. and Mendoza, G. A., Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1106.CrossRefGoogle Scholar
Farris, F., An intrinsic construction of Fefferman’s CR metric, Pacific J. Math. 123(1) (1986), 3345.CrossRefGoogle Scholar
Fefferman, C. L., Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), 395416.CrossRefGoogle Scholar
Fefferman, C. and Graham, C. R., “Conformal invariants”, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque, hors-série, Société mathématique de France, Paris, 1985, 95116.Google Scholar
Graham, C. R. and Hirachi, K., “The ambient obstruction tensor and Q-curvature”, in AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lectures in Mathematics and Theoretical Physics 8, (ed. Biquard, O.) European Mathematical Society, Université Louis Pasteur, Strasbourg, 2005, 5971.CrossRefGoogle Scholar
Graham, C. R., Jenne, R., Mason, L. J. and Sparling, G. A. J., Conformally invariant powers of the Laplacian, I: Existence, J. Lond. Math. Soc. (2) 46(3) (1992), 557565.CrossRefGoogle Scholar
Gursky, M. J. and Székelyhidi, G., A local existence result for Poincaré-Einstein metrics, preprint, 2017, arXiv:1712.04017.Google Scholar
Herzlich, M., A remark on renormalized volume and Euler characteristic for ACHE 4-manifolds, Differential Geom. Appl. 25(1) (2007), 7891.CrossRefGoogle Scholar
Herzlich, M., The canonical Cartan bundle and connection in CR geometry, Math. Proc. Cambridge Philos. Soc. 146 (2009), 415434.10.1017/S0305004108001527CrossRefGoogle Scholar
Jerison, D. and Lee, J. M., Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geometry 29 (1989), 303343.CrossRefGoogle Scholar
Lee, J. M., The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc. 296(1) (1986), 411429.Google Scholar
Lee, J. and Melrose, R., Boundary behaviour of the complex Monge–Ampère equation, Acta Math. 148 (1982), 159192.CrossRefGoogle Scholar
Marugame, T., Self-dual Einstein ACH metric and CR GJMS operators in dimension three, preprint, 2018, arXiv:1802.01264.10.2140/pjm.2019.301.519CrossRefGoogle Scholar
Matsumoto, Y., Asymptotics of ACH-Einstein metrics, J. Geom. Anal. 24 (2014), 21352185.CrossRefGoogle Scholar
Milnor, T. K., Harmonically immersed surfaces, J. Differential Geometry 14 (1979), 205214.CrossRefGoogle Scholar