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

Abstract

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small L 2 -norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an ε-regularity theorem, thus confirming a conjecture of Cheeger-Tian [20]. 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.

Counting ends on shrinkers

“…For the 4-dimensional case, Li and Wang [31] proved that any nontrivial flat cone cannot be approximated by smooth shrinkers with bounded scalar curvature and Harnack inequality under the pointed-Gromov-Hausdorff topology. Huang [22] applied the strategy of Cheeger-Tian [9] in Einstein manifolds and proved an -regularity theorem for 4-dimensional shrinkers, confirming a conjecture of Cheeger-Tian [9].…”

On the regular-convexity of Ricci shrinker limit spaces

“…It follows from Corollary 2.4 and an argument based on the original one of Cheeger-Colding's. For the sake of completeness we write down the technical details, see also [16] for a version only for Ricci shrinkers.…”

Global $\varepsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound

“…In [MuWa15], Munteanu-Wang proved that 4-dimensional shrinking Ricci solitons have bounded Riemann curvature provided a bounded scalar curvature condition, where the Riemann curvature bound depends on the local geometry around the base point. In [Hu20], Huang proved an ε-regularity for shrinking Ricci solitons with ε depending on the distance to the base point.…”