Ricci flow smoothing for locally collapsing manifolds

Abstract

We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

“…The same conclusion actually holds without the Ricci curvature upper bound, and this has been proven very recently by the authors based on the Ricci flow local smoothing techniques ([54, Theorem 1.4]); see Theorem 4.9 in the next section. Notice that the Assumption (1) in Theorem 2.9 is crucial in the original blow up argument, and the case of orbifold collapsing limit in [54,Theorem 1.4] is considerably more difficult; see Remark 3. Based on these results, the following theorem is recently proven in [53].…”

“…This lemma is a slight re-wording of [49,Lemma 1.11], which concerns non-collapsing initial data. See also [54,Lemma 4.1] for a proof.…”

The local entropy along Ricci flow Part A: the no-local-collapsing theorems

“…In the presence of the local isometry group actions, the diffeomorphisms constructed from the Ricci flows seem natural and respect the group actions. For further discussions, see [66], [67] and [64]. The conditions in Theorem 1.3 are not optimal and could be modified in various applications.…”

“…The bounded curvature condition can be dropped by exploiting the method of Hochard [57]. See also [56] and [67] for more details. Actually, we can find a sequence of cutoff functions 0 ≤ η k = e −2 f k ≤ 1 whose support set is U k = B(x 0 , 2k) for some fixed x 0 ∈ M and η k ≡ 1 on D k = B(x 0 , k).…”

“…Actually, we can find a sequence of cutoff functions 0 ≤ η k = e −2 f k ≤ 1 whose support set is U k = B(x 0 , 2k) for some fixed x 0 ∈ M and η k ≡ 1 on D k = B(x 0 , k). Moreover, we can ensure that η −1 k satisfies some uniform bounds(see (2.5) of [67]). Clearly, we have…”