L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

Guangwen Zhao

Paper · Jul 30, 2026

$L^{p}$ Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

Guangwen Zhao

math.DGarXiv:2607.28057v2

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Abstract

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M, g, J, f)$ and a real-valued pluriharmonic function $u$ , we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $\int_{M} ∣\nabla u ∣^{p} d v < \infty$ for some $0 < p < \infty$ . In the shrinking case, we prove the same conclusion for $0 < p \leq 2$ . Finally, we construct a complete Kähler example showing that the extension to the range $0 < p < 1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.

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Submitted: Jul 30, 2026
Last revised: Aug 5, 2026 (v2)
Subjects: math.DG

Comments: This version corrects a minor error in Lemma 3.3 as well as some typographical errors

Ricci solitons Kähler–Ricci solitons Kähler geometry

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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)

In the guidebook

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.

[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.

$L^{p}$ Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

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Lp Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

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$L^{p}$ Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons