Local classification of conformally-Einstein Kähler metrics in higher dimensions

Abstract

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real parameters along with an arbitrary Kähler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we call {\it special Kähler-Ricci potentials}.

38 pages, AMSTeX, submitted to the Proceedings of the London Mathematical Society

Cited by in corpus (12)

• Special Kähler-Ricci potentials on compact Kähler manifolds
• Warped product Kaehler manifolds and Bochner-Kaehler metrics
• Kahler manifolds with quasi-constant holomorphic curvature
• On cohomogeneity one Hermitian non-Kähler metrics
• Compact conformally Kahler Einstein-Weyl manifolds
• On weakly Einstein Kähler surfaces
• Generalized Calabi type Kahler surfaces
• Kähler manifolds with homothetic foliations by curves
• Special biconformal changes of Kähler surface metrics
• Special Ricci-Hessian equations on Kähler manifolds
• A detailed description of the generalized Calabi type Kahler surfaces
• Conformally-Kähler Ricci solitons and base metrics for warped product Ricci solitons