L^p Liouville type theorems for harmonic functions on gradient Ricci solitons

Yong Luo

Paper · Nov 7, 2023

$L^{p}$ Liouville type theorems for harmonic functions on gradient Ricci solitons

Yong Luo

math.DGarXiv:2311.03759v2

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Abstract

In this paper we consider $L^{p}$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M, g)$ is a gradient shrinking or steady Kähler-Ricci soliton, then we prove that any pluriharmonic function $u$ on $M$ with $\nabla u \in L^{p} (M)$ for some $1 < p \leq 2$ is a constant function.

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Submitted: Nov 7, 2023
Last revised: Nov 27, 2024 (v2)
Subjects: math.DG

Comments: Errors corrected, main theorems weakened, and add Theorem 1.3; 18 pages

Ricci solitons Kähler–Ricci solitons Kähler geometry

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Background reading

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 type theorems for harmonic functions on gradient Ricci solitons

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Lp Liouville type theorems for harmonic functions on gradient Ricci solitons

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$L^{p}$ Liouville type theorems for harmonic functions on gradient Ricci solitons