Tohoku Mathematical Journal

A non-existence theorem of proper harmonic morphisms from weakly asymptotically hyperbolic manifolds

Xiaohuan Mo and Yuguang Shi

Full-text: Open access

Abstract

We prove a non-existence theorem for proper harmonic morphisms from weakly asymptotically hyperbolic manifolds to hyperbolic manifolds which are $C^2$ up to the boundary at infinity.

Article information

Source
Tohoku Math. J. (2), Volume 58, Number 3 (2006), 359-368.

Dates
First available in Project Euclid: 17 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1163775135

Digital Object Identifier
doi:10.2748/tmj/1163775135

Mathematical Reviews number (MathSciNet)
MR2273275

Zentralblatt MATH identifier
1119.53047

Subjects
Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 58E20: Harmonic maps [See also 53C43], etc.

Keywords
Harmonic morphism weakly asymptotically hyperbolic manifold conformally compact manifold proper map harmonic map

Citation

Mo, Xiaohuan; Shi, Yuguang. A non-existence theorem of proper harmonic morphisms from weakly asymptotically hyperbolic manifolds. Tohoku Math. J. (2) 58 (2006), no. 3, 359--368. doi:10.2748/tmj/1163775135. https://projecteuclid.org/euclid.tmj/1163775135


Export citation

References

  • L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperolic manifolds, Ann. Global Anal. Geom. 16 (1998), 1--27.
  • P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monogr. (N.S.), Oxford University Press, 2003.
  • P. T. Chruściel, J. Jezierski and S. \Lęski, The Trautman-Bondi mass of hyperboloidal initial data sets, Adv. Theor. Math. Phys. 8(2004), 83--139.
  • B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107--144.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1983.
  • T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto. Univ. 19 (1979), 215--229.
  • P. Li and L. F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1--46.
  • P. Li and L. F. Tam, Uniquess and regularity of proper harmonic maps, Ann of Math. (2) 137 (1993), 167--201.
  • P. Li and J. Wang, Convex hull properties of harmonic maps, J. Differential Geom. 48 (1998), 497--530.
  • R. Mazzeo, The Hodge conformally of a compact metric, J. Differential Geom. 28 (1988), 309--339.
  • X. Mo and Y. Shi, A nonexistence theorem of proper harmonic morphisms between hyperbolic spaces, Geom. Dedicata 93 (2002), 89--94.
  • Y. Shi, L. F. Tam and T. Y. Wan, Harmonic maps on hyperbolic spaces with singular boundary value, J. Differential Geom. 51 (1999), 551--600.
  • T. Wan and Y. Xin, Vanishing theorems for conformally compact manifolds, Comm. Partal Differential Equations 29 (2004), 1267--1279.
  • X. Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), 273--299.
  • E. Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3(1999), 1635--1655.
  • J. C. Wood, The geometry of harmonic maps and morphisms, In Proceedings of the fourth international workshop on differential geometry and its applications, (Brasov, Romania, 1999), 306--313, Transilvania University Press, 1999.