Classification of gradient Einstein-type Kähler manifolds with \(\alpha =0\)

Thanks to an ambitious project initiated by Catino, Mastrolia, Monticelli, and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, k-Yamabe solitons, quasi-Yamabe solitons, and conformal solitons, can now be studied under a unified framework known as Einstein-type manifolds. Einstein-type manifolds are characterized by four constants, denoted by \(\alpha , \beta , \mu \), and \(\rho \). In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with \(\alpha = 0\). As a corollary, we obtain rotational symmetry for many classes. In particular, we show that any non-trivial complete quasi-Yamabe gradient soliton on Kähler manifolds is rotationally symmetric.