Abstract
Let ϕ be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to ϕ, derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to ϕ being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.
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Baird, P.: Harmonic morphisms and circle actions on 3- and 4-manifolds. Ann. Inst. Fourier 40 (1990), 177–212.
Baird, P.; Eells, J.: A cconservation law for harmonic maps. Geomtery Symp. Utrecht 1980, Proceedings, Lecture Notes in Math. 894, Springer-Verlag, 1981, 1–25.
Baird, P., Wood, J.C.: Bernstein theorems for harmonic morphisms from R3 and S3. Math. Ann. 280(1988), 579–603.
Baird, P.; Wood, J. C.: Harmonic morphisms, Seifert fibre spaces and conformal foliations. To appear in: Proc. London Math. Soc..
Eells, J.; Ratto, A.: Harmonic maps between spheres and ellipsoids. Internat. J. Math. 1 (1990), 1–27.
Fuglede, B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier 28(1978), 107–144.
Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19 (1979), 215–229.
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Baird, P. A Böchner technique for harmonic mappings from a 3-manifold to a surface. Ann Glob Anal Geom 10, 63–72 (1992). https://doi.org/10.1007/BF00128338
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DOI: https://doi.org/10.1007/BF00128338