Vaisman manifolds and transversally Kähler–Einstein metrics

Abstract

We use the transverse Kähler–Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse Kähler–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 Kähler–Ricci flow on Vaisman manifolds, recovering in a particular setting a result of Bedulli et al. (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.