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

Abstract 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 $$\alpha , \beta , \mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> , and $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> . In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with $$\alpha = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.