A gap theorem of four-dimensional gradient shrinking solitons

2016/06/03 by Zhuhong Zhang, Zhang, Zhuhong · 1 citation
Mathematics · Physics and Astronomy · #53C25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1606.01154

openalex publication_date 2016/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we will prove a gap theorem for four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or λ₁ + λ₂≥ c₀ R>0 everywhere for some c₀≈ 0.29167, where \λ_i\ are the two least eigenvalues of Ricci curvature. Furthermore, we will show that λ₁ + λ₂≥ \frac 13R>0 under a better pinched Weyl tensor assumption. We point out that the lower bound \frac 13R is sharp.

Citations

• Gradient shrinking Ricci solitons of half harmonic Weyl curvature
• Geometry of shrinking Ricci solitons

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• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

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