Tohoku Mathematical Journal

Biharmonic maps and morphisms from conformal mappings

Eric Loubeau and Ye-Lin Ou

Full-text: Open access

Abstract

Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e., non-harmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.

Article information

Source
Tohoku Math. J. (2), Volume 62, Number 1 (2010), 55-73.

Dates
First available in Project Euclid: 31 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1270041027

Digital Object Identifier
doi:10.2748/tmj/1270041027

Mathematical Reviews number (MathSciNet)
MR2654303

Zentralblatt MATH identifier
1202.53061

Subjects
Primary: 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Keywords
Biharmonic maps conformal maps biharmonic morphisms

Citation

Loubeau, Eric; Ou, Ye-Lin. Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. (2) 62 (2010), no. 1, 55--73. doi:10.2748/tmj/1270041027. https://projecteuclid.org/euclid.tmj/1270041027


Export citation

References

  • S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. InterEditions, Editions du C.N.R.S., Meudon, 1991.
  • P. Baird, A. Fardoun and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Global Anal. Geom. 34 (2008), 403--414.
  • P. Baird and S. Gudmundsson, $p$-Harmonic maps and minimal submanifolds, Math. Ann. 294 (1992), 611--624.
  • P. Baird and D. Kamissoko, On constructing biharmonic maps and metrics, Ann. Global Anal. Geom. 23 (2003), 65--75.
  • P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, Oxford Science Publications, 2003.
  • A. Balmus, Biharmonic properties and conformal changes, An. Ştiinţ. Univ. Al. I. Cuza IaŞi. Mat. (N.S.) 50 (2004), 361--372.
  • S. Y. A. Chang, L. Wang and P. C. Yang, A regularity theory of biharmonic maps, Comm. Pure Appl. Math. 52 (1999), 1113--1137.
  • D. M. DeTurck and J. L. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. 14 (1981), 249--260.
  • F. Duzaar and M. Fuchs, Existence and regularity of functions which minimize certain energies in homotopy classes of mappings, Asymptotic Anal. 5 (1991), 129--144.
  • J. Eells and L. Lemaire, Selected topics in harmonic maps. CBMS Regional Conf. Ser. in Math. 50, American Mathematical Society, Providence, RI, 1983.
  • B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1979), 107--144.
  • B. Fuglede, Harmonic morphisms between semi-Riemannian manifolds, Ann. Acad. Sci. Fenn. Math. 21 (1996), 31--50.
  • R. Hardt and F. H. Lin, Mappings minimizing the $L\sp p$ norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 555--588.
  • T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215--229.
  • G. Y. Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), 389--402.
  • J. Jost, A conformally invariant variational problem for mappings between Riemannian manifolds, Preprint, Centre for Mathematical Analysis, ANU, 1984.
  • E. Loubeau and Y.-L. Ou, The characterization of biharmonic morphisms, Differential geometry and its applications (Opava 2001), 31--42, Math. Publ. 3, Silesian Univ. Opava, Opava, 2001.
  • Y.-L. Ou, Biharmonic morphisms between Riemannian manifolds, Geometry and topology of submanifolds, X (Beijing/Berlin, 1999), 231--239, World Sci. Publ., River Edge, N.J., 2000.
  • Y.-L. Ou, $p$-harmonic morphisms, biharmonic morphisms and nonharmonic biharmonic maps, J. Geom. Phys. 56 (2006), 358--374.
  • C. Wang, Stationary biharmonic maps from $\boldsymbolR\sp m$ into a Riemannian manifold, Comm. Pure Appl. Math. 57 (2004), 419--444.
  • C. Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations 21 (2004), 221--242.
  • C. Wang, Biharmonic maps from $\boldsymbolR\sp 4$ into a Riemannian manifold, Math. Z. 247 (2004), 65--87.
  • S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their application to geometry, Indiana Univ. Math. J. 25 (1976), 659--670.