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.

The Stability Inequality for Ricci-Flat Cones

“…One reason for this is that the integrals |A| 2 dµ and |Rm| 2 dV are scale invariant in dimension 2 and 4, respectively. These integrals show up naturally in the compactness theorems for shrinkers in [14] and [31] and can be estimated using the Gauss-Bonnet formula in the respective dimension. As hinted at in [8], there is hope that actually all but very few singularity models for four-dimensional Ricci flow are unstable (the known stable examples are S 4 , S 3 ×R, S 2 ×R 2 , CP 2 and their quotients, and presumably the blow-down shrinker of Feldman-Ilmanen-Knopf [20] is also stable).…”

A gap theorem on complete shrinking gradient Ricci solitons

“…(3) in the above lemma is proved by Cao-Zhou [2](see also Fang-Man-Zhang [9] and for an improvement, Haslhofer-Müller [12]). It is well known that a complete shrinking gradient Ricci solitons has nonnegative scalar curvature (see Chen [4]) and either R > 0 or the metric g is flat (see Pigola-Rimoldi-Setti [17] or the author [21]).…”

ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons

“…This was first proved by Cao-Zhou [6]. Here, we shall need the improved version by Haslhofer-Müller [33]:…”

“…Anderson's ε-regularity with respect to collapsing [1] is the starting point of Cheeger-Tian's ε-regularity theorem for four dimensional Einstein manifolds [20]. By the bound on the Sobolev constant for dµ f , as obtained in Lemma 2.9, the proof of this theorem is by now standard using Moser iteration, see [1] and [33] for the original work.…”