12 research outputs found
A NOTE ON THE COMPACTNESS THEOREM FOR 4d RICCI SHRINKERS
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem
for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper
bound for the Euler characterisic, and a lower bound for the gradient of the
potential at large distances. In this note, we show that the last two
assumptions in fact can be removed. The key ingredient is a recent estimate of
Cheeger-Naber arXiv:1406.6534
-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons
A closed four dimensional manifold cannot possess a non-flat Ricci soliton
metric with arbitrarily small -norm of the curvature. In this paper, we
localize this fact in the case of shrinking Ricci solitons by proving an
-regularity theorem, thus confirming a conjecture of Cheeger-Tian.
As applications, we will also derive structural results concerning the
degeneration of the metrics on a family of complete non-compact four
dimensional shrinking Ricci solitons without a uniform entropy lower bound. In
the appendix, we provide a detailed account of the equivariant good chopping
theorem when collapsing with locally bounded curvature happens
Rigidity and {\epsilon}-regularity theorems of Ricci shrinkers
In this paper, we study the rigidity and {\epsilon}-regularity theorems of
Ricci shrinkers. First we prove the rigidity of the asymptotic volume ratio and
local volume around a base point of a non-compact Ricci shrinker. Next we
obtain some {\epsilon}-regularity theorems of local entropy and curvature,
which improve the previous corresponding results essentially and use them to
study the structure of Ricci shrinkers at infinity. Especially, if the
curvature of a non-compact Ricci shrinker satisfies some natural integral
conditions, then it is asymptotic to a cone.Comment: 31 paper