On gradient Ricci solitons

2009/10/06 by Ovidiu Munteanu, Munteanu, Ovidiu, Nataša Šešum +1 · 11 citations
Mathematics · Physics and Astronomy · #53C44 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0910.1105

openalex publication_date 2009/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons and show that if the total energy of a harmonic function on such a manifold is finite then the function is constant. Consequences to the structure of the manifold at infinity are also discussed.

Citations

• On Locally Conformally Flat Gradient Shrinking Ricci Solitons
• Ricci solitons: the equation point of view
• Noncompact Shrinking 4-Solitons with Nonnegative Curvature

Cited by

• Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas
• The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
• On curvature estimates for four-dimensional gradient Ricci solitons
• Local Structure of Gradient Almost Ricci Solitons with Harmonic Weyl Tensor
• Topology at infinity of complete gradient Schouten solitons
• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons
• Ricci-Yamabe solitons on a Walker 3-manifold
• Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor
• Four-dimensional shrinkers with nonnegative Ricci curvature
• Ricci Solitons on the Poincaré upper half plane

Related

• Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor
• The Curvature of Gradient Ricci Solitons
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature
• Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors
• On the rigidity of harmonic-Ricci solitons