MULTIPLIER HERMITIAN-EINSTEIN METRICS ON FANO MANIFOLDS OF KSM-TYPE

In this article we focus on multiplier Hermitian -Einstein metrics introduced by Mabuchi which include K & auml;hler-Einstein metrics, K & auml;hler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are introduced by the first author as toric bundles, to establish a criterion for the existence of multiplier Hermitian -Einstein metrics in terms of KSM-data. An explicit example for a KSM-manifold admitting a family of multiplier Hermitian -Einstein metrics is constructed by using a continuous path connecting a K & auml;hler-Ricci soliton with a Mabuchi soliton.