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Register a new userWeierstrass representations for harmonic morphisms on Euclidean spaces and spheres
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Authors
P. Baird
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We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.
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We give a classification of quadratic harmonic morphisms between Euclidean spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford systems. In the case $ {Bbb R}^{4}longrightarrow {Bbb R}^{3} $, we determine all quadratic harmonic morphisms and show that, up to a constant factor, they are all bi-equivalent (Definition 3.2) to the well-known Hopf construction map and induce harmonic morphisms bi-equivalent to the Hopf fibration ${Bbb S}^{3} longrightarrow {Bbb S}^{2}$.
P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all harmonic morphisms from any three-dimensional Euclidean and spherical space form to a surface, namely that any such harmonic morphism is the composition of a standard harmonic morphism and a weakly conformal map. We list the standard harmonic morphisms.
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.
We provide explicit, simple, geometric formulas for free involutions rho of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S^n/rho is a manifold that is homotopically equivalent but not diffeomorphic to RP^n. We use these formulas for constructing explicit non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13) and to provide explicit formulas for non-cancellation phenomena in group actions.
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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