Geometry of shrinking Ricci solitons

2015/07/28 by Ovidiu Munteanu, Jiaping Wang · 10 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.1112/s0010437x15007496

Abstract

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton M with bounded scalar curvature S , it is shown that the curvature operator Rm of M satisfies the estimate |Rm|\leqslant cS for some constant c . Moreover, the curvature operator Rm is asymptotically nonnegative at infinity and admits a lower bound Rm\geqslant -c(ln (r+1))^-1/4 , where r is the distance function to a fixed point in M . As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

Cited by

• Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons
• Conical structure for shrinking Ricci solitons
• Curvature estimates for steady and expanding solitons in higher dimensions
• Geometry of Two-dimensional Self-shrinkers
• Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature
• On curvature estimates for four-dimensional gradient Ricci solitons
• Curvature pinching of asymptotically conical gradient expanding Ricci solitons
• Topology at infinity of complete gradient Schouten solitons
• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• A gap theorem of four-dimensional gradient shrinking solitons
• A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons
• Curvature estimates for steady Ricci solitons
• Geometric Structure of Ends of Ricci Shrinkers

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• The Curvature of Gradient Ricci Solitons
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• Shrinking Ricci solitons with positive isotropic curvature