Restrictions on harmonic morphisms

Mustafa, M.

doi:10.1090/S1088-4173-99-00026-0

Restrictions on harmonic morphisms
HTML articles powered by AMS MathViewer

by M. T. Mustafa

Conform. Geom. Dyn. 3 (1999), 102-115

DOI: https://doi.org/10.1090/S1088-4173-99-00026-0

Published electronically: August 16, 1999

PDF | Request permission

Abstract:

We consider horizontally (weakly) conformal maps $\phi$ between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of $\phi$, using the language of moving frames. Applying this formula to harmonic horizontally (weakly) conformal maps or equivalently to harmonic morphisms we obtain a Weitzenböck formula similar to an earlier result of the author (J. London Math. Soc. (2) 57 (1998), 746–756), and hence vanishing results for harmonic morphisms from compact manifolds of positive curvature. Further, a method is developed to obtain restrictions on harmonic morphisms from some non-compact and non-positively curved domains. Finally, a discussion of restrictions on harmonic morphisms between simply connected space forms is given.

References

Paul Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Mathematics, vol. 87, Pitman (Advanced Publishing Program), Boston, MA, 1983. MR 716320
Paul Baird and John C. Wood, Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms, J. Austral. Math. Soc. Ser. A 51 (1991), no. 1, 118–153. MR 1119693, DOI 10.1017/S1446788700033358
Bryant, R. L.: Harmonic morphisms with fibres of dimension one, Communications in Analysis and Geometry (to appear).
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385–524. MR 956352, DOI 10.1112/blms/20.5.385
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
James Eells and Paul Yiu, Polynomial harmonic morphisms between Euclidean spheres, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2921–2925. MR 1273489, DOI 10.1090/S0002-9939-1995-1273489-4
Bent Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 2, vi, 107–144 (English, with French summary). MR 499588
S. I. Goldberg, T. Ishihara, and N. C. Petridis, Mappings of bounded dilatation of Riemannian manifolds, J. Differential Geometry 10 (1975), no. 4, 619–630. MR 390964, DOI 10.4310/jdg/1214433165
Sigmundur Gudmundsson, Harmonic morphisms between spaces of constant curvature, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 133–143. MR 1200192, DOI 10.1017/S0013091500005940
Sigmundur Gudmundsson, Harmonic morphisms from complex projective spaces, Geom. Dedicata 53 (1994), no. 2, 155–161. MR 1307290, DOI 10.1007/BF01264019
Gudmundsson, S.: The Bibliography of Harmonic Morphisms, http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html.
Tôru Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), no. 2, 215–229. MR 545705, DOI 10.1215/kjm/1250522428
Atsushi Kasue and Takumi Washio, Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math. 27 (1990), no. 4, 899–928. MR 1088189
Joachim Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655–683. MR 1307899, DOI 10.2307/2118620
Montaldo, S.: Harmonic maps and morphisms via moving frames, Lecture notes, University of Leeds (1997).
Mustafa, M. T.: A Bochner technique for harmonic morphisms, J. London Math. Soc. (2) 57 (1998) 746–756.
Mustafa, M. T. and Wood, J. C.: Harmonic morphisms from three-dimensional Euclidean and spherical space forms, Algebras, Groups and Geometries 15 (1998) 155–172.
Ye-Lin Ou and John C. Wood, On the classification of quadratic harmonic morphisms between Euclidean spaces, Algebras Groups Geom. 13 (1996), no. 1, 41–53. MR 1392037
T. J. Willmore, Riemannian geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. MR 1261641
J. C. Wood, Harmonic morphisms, foliations and Gauss maps, Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984) Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 145–184. MR 833811, DOI 10.1090/conm/049/833811
John C. Wood, Harmonic maps and morphisms in $4$ dimensions, Geometry, topology and physics (Campinas, 1996) de Gruyter, Berlin, 1997, pp. 317–333. MR 1605260