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Register a new userHarmonic morphisms with one-dimensional fibres on Einstein manifolds
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We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.
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We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equation, and also the Beltrami fields equation of hydrodynamics, and lead to constructions of self-dual metrics.
We characterise the actions, by holomorphic isometries on a Kahler manifold with zero first Betti number, of an abelian Lie group of dimgeq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kahler moment map $phi$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kahler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $phi$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kahler structure, and we obtain an explicit global formula for the map. If dim Tgeq 2 and either $phi$ has critical points, or M is nonflat and dim M=4 dim T then $phi$ cannot be horizontally weakly conformal.
For a homotopically energy-minimizing map $u: N^3to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}{theta}$ to the scalar curvature of $N$ and the mean curvature of the boundary $partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowkas characterization for irreducible manifolds to arbitrary topologies.
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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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