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Register a new userHarmonic morphisms between almost Hermitian manifolds
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Authors
S. Gudmundsson
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We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian structure on its domain.
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In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canonical second fundamental form, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, and Simons integral inequality of pseudoholomorphic isometric immersion.
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Equivalences between conformal foliations on Euclidean $3$-space, Hermitian structures on Euclidean $4$-space, shear-free ray congruences on Minkowski $4$-space, and holomorphic foliations on complex $4$-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued harmonic morphism without critical points defined on an open subset of Minkowski space is conformally equivalent to the direction vector field of a shear-free ray congruence, 2) the boundary values at infinity of a complex-valued harmonic morphism on hyperbolic $4$-space define a real-analytic conformal foliation by curves of an open subset of Euclidean $3$-space and all such foliations arise this way. This gives an explicit method of finding such foliations; some examples are given.
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