A compactness theorem for complete Ricci shrinkers

Abstract

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.

28 pages, final version, to appear in GAFA

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Cited by in corpus (36)

• Geometry of shrinking Ricci solitons
• Compactness theory of the space of Super Ricci flows
• Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
• Structure theory of non-collapsed limits of Ricci flows
• $\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons
• A lower bound for the scalar curvature of noncompact nonflat Ricci shrinkers
• Geometry of manifolds with densities
• Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers
• Heat kernel on Ricci shrinkers
• Ancient solutions to the Kähler Ricci flow
• The Curvature of Gradient Ricci Solitons
• Liouville-type theorems on the complete gradient shrinking Ricci solitons
• Small curvature concentration and Ricci flow smoothing
• Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers
• Positive solutions to Schrödinger equations and geometric applications
• Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons
• Time analyticity for heat equation on gradient shrinking Ricci solitons
• A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons
• On the regular-convexity of Ricci shrinker limit spaces
• Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons
• Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature
• Rigidity of the round cylinders in Ricci shrinkers
• A bound for the order of the fundamental group of a complete noncompact Ricci shrinker
• On kappa-noncollapsed complete noncompact shrinking gradient Ricci solitons which split at infinity
• A rigidity theorem for codimension one shrinking gradient Ricci solitons in ${\mathbb{R}}^{n+1}$
• The sharp Li-Yau equality on Shrinking Ricci Solitons with applications
• On the structure of Ricci shrinkers
• The stability inequality for Ricci-flat cones
• On Ricci Solitons with Isoparametric Potential Functions
• Ends of (singular) Ricci shrinkers
• The weighted ambient metric for manifolds with density
• The Rigidity of Ricci shrinkers of dimension four
• A compactness theorem for four-dimensional shrinking gradient Ricci solitons
• A gap theorem on complete shrinking gradient Ricci solitons
• $\epsilon$-regularity for shrinking Ricci solitons and Ricci flows
• Sharp upper diameter bounds for compact shrinking Ricci solitons