Generalized Quasi-Einstein metrics on admissible manifolds

TL;DR

This paper proves the existence of Generalized Quasi-Einstein K"ahler metrics on certain admissible manifolds derived from base manifolds with constant scalar curvature, highlighting conditions for their existence in various K"ahler classes.

Contribution

It establishes existence results for Generalized Quasi-Einstein metrics on admissible manifolds and provides examples illustrating when such metrics do or do not exist.

Findings

Existence of Generalized Quasi-Einstein metrics in small admissible K"ahler classes.

Counterexamples where existence fails in some classes.

Extension of results to metrics with scalar curvature as an affine combination of a Killing potential and its Laplacian.

Abstract

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and T{\o}nnesen-Friedman), arising from a base with a local K\"ahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein K\"ahler metrics (as defined by D. Guan) in all "sufficiently small" admissible K\"ahler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some K\"ahler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature is an affine combination of a Killing potential and its Laplacian.