The rigidity of Ricci shrinkers of dimension four

Abstract

In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.Ricci shrinker is also called as gradient shrinking Ricci soliton. Direct calculation shows that ∇(R + |∇ f | 2 − f ) = 0. Adding f by a constant if necessary, we assume throughout that3) where µ = µ(g, 1) is the entropy functional of Perelman(c.f. [35]). The Ricci shrinker (1.1) was introduced by Hamilton [22] in mid 1980's. As critical points of Perelman's µ-entropy, the Ricci shrinkers play important roles in the singularity analysis of the Ricci flow. For example, it is proved by Enders-Müller-Topping [20] that the proper rescaling limit of a type-I singularity is always a nontrivial Ricci shrinker. More information and references can be found in Chapter 30 of the book [18] by Chow and his coauthors. In dimension 2, the only Ricci shrinkers are R 2 , S 2 and RP 2 with standard metrics, due to the classification of Hamilton [23]. In dimension 3, based on the breakthrough of Perelman([35], [36]), through the efforts of Naber [33], Ni-Wallach [34] and Cao-Chen-Zhu [6], etc, we know that R 3 , S 2 × R, S 3 and their quotients are all the possible Ricci shrinkers. In dimension 4 and higher, much fewer is known about Ricci shrinkers. Typically, some extra conditions of the curvature operator(e.g. [3], [9], [7], [17], etc), or geometric properties at infinity(c.f. [27], [32]) are required to draw definite geometry conclusion. We refer the readers the surveys [4], [5] for more detailed picture. Without such extra conditions, it is still not clear how to classify Ricci shrinkers. However, one can study the moduli of Ricci shrinkers. In [8], Cao-Sesum showed the weak compactness of the moduli of Kähler Ricci solitons with uniformly bounded diameter and uniformly lower bound of Ricci curvature and µ-functional. The extra conditions were gradually weakened or removed by X. Zhang [39], Weber [38], Chen-Wang [15], Z.L. Zhang [40] and Haslhofer-Müller [24], etc.In this article, we focus on the study of the moduli M * (A, H), which consists of 4d Ricci shrinkers which have uniform entropy lower bound and Harnack inequality of scalar curvature on unit ball(c.f. Definition 2.3 for precise definition). Moreover, we require each Ricci shrinker has bounded(not uniformly) scalar curvature. In light of the results of Haslhofer-Müller(c.f. Theorem 2.6), it is known that such moduli has weak compactness. In other words, any sequence of Ricci shrinkers in M * (A, H) sub-converges to an orbifold Ricci shrinker with locally finite singularities, in the pointed-Gromov-Hausdorff topology. Now we reverse the process and ask the following question:What kind of orbifolds can be approximated by a sequence of Ricci shrinkers in M * (A, H)?Note that flat cones R 4 /Γ are naturally the s...

Counting ends on shrinkers

“…Li, Li and Wang [26] gave a structure theory for non-collapsed shrinkers, which was further developed by Huang, Li and Wang [23]. 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].…”

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

“…If this conjecture is confirmed, the task of classifying 4-d Ricci shrinkers will be reduced to the classification with a given Euler characteristic bound, with the help of the resulting uniform entropy lower bound. See [37] for several uniform estimates in this situation.…”

Heat kernel on Ricci shrinkers

“…It was proved in [28, Theorem 1] that µ is the optimal log-Sobolev constant for all scales. Following [27], we have the following definition. (2.2)…”