Geometry of shrinking Ricci solitons

Abstract

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le cS$ for some constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\mathrm{Rm}\ge -c{\left(\ln r\right)}^{-1/4},$ where $r$ is the distance function to a fixed point in $M$. As application, we prove that if the scalar curvature converges to zero at infinity, then the manifold 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.

28 pages, submitted, v2 has a new section about the conical structure of solitons

References in corpus (1)

• A lower bound for the scalar curvature of noncompact nonflat Ricci shrinkers

Cited by in corpus (28)

• Classification of expanding and steady Ricci solitons with integral curvature decay
• Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
• $\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons
• Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers
• Curvature estimates for four-dimensional complete gradient expanding Ricci solitons
• Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature
• Linear stability of compact shrinking Ricci solitons
• Geometry of Two-dimensional Self-shrinkers
• Some curvature pinching results for Riemannian manifolds with density
• Conical structure for shrinking Ricci solitons
• Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons
• Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio
• Four-dimensional complete gradient shrinking Ricci solitons
• Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones
• Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature
• Positive solutions to Schrödinger equations and geometric applications
• A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons
• On curvature estimates for four-dimensional gradient Ricci solitons
• Isometries of asymptotically conical shrinking Ricci solitons
• Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers
• A gap theorem of four-dimensional gradient shrinking solitons
• Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons
• Sharp upper diameter bounds for compact shrinking Ricci solitons
• Global $\epsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound
• The Rigidity of Ricci shrinkers of dimension four
• Curvature estimates for steady Ricci solitons
• Ends of (singular) Ricci shrinkers
• $\epsilon$-regularity for shrinking Ricci solitons and Ricci flows