Special Ricci-Hessian equations on Kähler manifolds

Andrzej Derdzinski; Paolo Piccione

Paper · Nov 2, 2023

Special Ricci-Hessian equations on Kähler manifolds

Andrzej Derdzinski, Paolo Piccione

math.DGarXiv:2311.01345v3

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Abstract

Special Ricci-Hessian equations on Kähler manifolds $(M, g)$ , as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367–380] involve functions $τ$ on $M$ and state that, for some function $α$ of the real variable $τ$ , the sum of $α \nabla d τ$ and the Ricci tensor equals a functional multiple of the metric $g$ , while $α \nabla d τ$ itself is assumed to be nonzero almost everywhere. Three well-known obvious "standard" cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $α = 2 cot τ$ , or $α = 2 coth τ$ , or $α = 2 tanh τ$ . We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a "nonstandard" way.

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Submitted: Nov 2, 2023
Last revised: Jul 28, 2024 (v3)
Subjects: math.DG
Journal: Journal of the Australian Mathematical Society, vol. 120 (2026), no. 1, pp. 57-77
DOI: 10.1017/S1446788725000102

Comments: Proof of Theorem E simplified

Ricci solitons Kähler–Ricci solitons Kähler geometry

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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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 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)

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Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
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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.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 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.

Special Ricci-Hessian equations on Kähler manifolds

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