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Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation

Liviu Ornea·Misha Verbitsky
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Abstract

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its volume form. This result is a conformal analogue of Calabi's theorem stating the uniqueness of Kaehler metrics with a given volume form in a given Kaehler class. We prove that a solution of a conformal version of complex Monge-Ampere equation is unique. We conjecture that a Hermitian Einstein-Weyl structure on a compact complex manifold is unique, up to a holomorphic automorphism, and compare this conjecture to Bando-Mabuchi theorem.

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Liviu Ornea, Misha Verbitsky. 2008-08-01. Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation. https://arxiv.org/abs/math/0606309

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