Vaisman manifolds and transversally K\"ahler-Einstein metrics

TL;DR

This paper explores the deformation of Vaisman metrics into transverse K"ahler-Einstein metrics using the transverse K"ahler-Ricci flow, providing new proofs and insights into Einstein-Weyl structures on Vaisman manifolds.

Contribution

It introduces a new approach to deform Vaisman metrics into K"ahler-Einstein metrics via the transverse K"ahler-Ricci flow without relying on Molino's theorem.

Findings

Established short time existence of transverse K"ahler-Ricci flow on Vaisman manifolds.

Connected Einstein-Weyl structures with quasi-Einstein metrics on Vaisman manifolds.

Provided examples illustrating the deformation process and geometric structures.

Abstract

We use the transverse K\"ahler-Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse K\"ahler-Einstein structure. We also study the main features of such a manifold. Among other results, using techniques from the theory of parabolic equations, we obtain a direct proof for the short time existence of the solution for transverse {\K}-Ricci flow on Vaisman manifolds, recovering in a particular setting a result of Bedulli, He and Vezzoni [J. Geom. Anal. 28, 697--725 (2018)], but without employing the Molino structure theorem. Moreover, we investigate Einstein-Weyl structures in the setting of Vaisman manifolds and find their relationship with quasi-Einstein metrics. Some examples are also provided to illustrate the main results.