Classification of gradient Einstein-type Kähler manifolds with $α=0$
Shun Maeta
TL;DR
The paper classifies all non-trivial, complete gradient Einstein-type Kähler manifolds with $α=0$, extending the unified Einstein-type framework of CMMR17 to the Kähler setting. A case-by-case analysis based on the signs and vanishing of $(β, μ)$ yields four families: (I) $β=0$ gives no non-trivial solutions; (II) $μ=0$ yields either M = R × N^{2m-1} with g = dr^2 + a^2 g_N and F(r) = a r + b, or complex Euclidean with F(r) = a r^2 + b; (III) $β$ and $μ$ opposite signs gives no non-trivial; (IV) $β$ and $μ$ same sign yields M = [0, +∞) × S^{2m-1} with g = dr^2 + ψ^2(r) g_S and ψ(r) = F'(r) e^{-c F(r)} where c = - μ/β. The analysis also yields explicit forms for the potential F and the warping function in these warped-product structures, and the rotational-symmetry corollaries show that many solitons, including every non-trivial complete gradient quasi-Yamabe soliton on a Kähler manifold, are rotationally symmetric.
Abstract
Thanks to the 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 $α, β, μ$ and $ρ$. In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with $α= 0$. As a corollary, rotational symmetry for many classes is obtained. In particular, we show that any non-trivial complete gradient quasi-Yamabe soliton on Kähler manifolds is rotationally symmetric.