L. Ornea and M. Verbitsky: Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampère equation, p.339-353

Abstract:

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kähler covering $\tilde M$, with the deck group acting on $\tilde M$ 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 Kähler metrics with a given volume form in a given Kähler class. We prove that the solution of the conformal version of complex Monge-Ampère 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.

Key Words: Einstein-Weyl structure, Vaisman manifold, potential.

2000 Mathematics Subject Classification: Primary: 53C55.

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