Paper · Mar 25, 2025

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

math.DGarXiv:2503.19596

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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.

Comments: 9 pages

Ricci solitons Almost & η-Ricci solitons Kähler geometry Rotational symmetry & warped products

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

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

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Classification of gradient Einstein-type Kähler manifolds with α=0

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