On the structure of Ricci shrinkers

Abstract

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.

Sharp upper diameter bounds for compact shrinking Ricci solitons

“…First, we apply a sharp logarithmic Sobolev inequality and a proper cut-off function to get a new functional inequality, which is related to the maximal function and the volume ratio (see Theorem 2.4). We point out that the sharp logarithmic Sobolev inequality is a key inequality in our paper, which was proved by Li, Li and Wang [29] for the compact Ricci solitons and then extended by Li and Wang [30] to the non-compact case. Second, we use the functional inequality to give an alternative theorem, which states that the maximal function of scalar curvature and the volume ratio cannot be simultaneously smaller than a fixed constant on a ball of shrinking Ricci soliton (see Theorem 3.1).…”

“…In the whole paper, we let a triple (M, g, f ) denote an n-dimensional gradient shrinking Ricci soliton. As in [29], normalizing f by adding a constant in (1.1), without loss of generality, we can assume gradient shrinking Ricci soliton (1.1) simultaneously satisfies…”

“…Note that µ is a finite constant for a fixed compact gradient shrinking Ricci soliton. From Lemma 2.5 in [29], we know that e µ is almost equivalent to the volume of the geodesic ball B(p 0 , 1), whose radius 1 centered at p 0 . To be more precise,…”

Geometric inequalities and rigidity of gradient shrinking Ricci solitons

“…where R is the scalar curvature of (M, g) and µ = µ(g, 1) is the entropy functional of Perelman [46]; see also the detailed explanation in [37] or [55,56]. For a compact shrinker, µ has a lower bound; but for the non-compact case, we generally need to assume µ > −∞ so that our discussion makes sense.…”

“…In particular, for the Euclidean space, we have µ = 0. In [37], Li, Li and Wang proved that e µ is nearly equivalent to V (p 0 , 1), i.e., the volume of geodesic ball B(p 0 , 1) centered at point p 0 ∈ M and radius 1. Here p 0 ∈ M is a point where f attains its infimum, which always exists on shrinkers but possibly is not unqiue; see [29].…”