Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

Abstract

In the study of collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem ([Fu1], [CFG]), and the stability for isometric compact Lie group actions on manifolds ([Pa], [GK]). The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry. IntroductionsIn the study of collapsed manifolds with bounded sectional curvature (| sec | ≤ 1), the following two results provide basic tools: a (singular) fibration theorem ([Fu1], [CFG]), and the stability for isometric compact Lie group actions on manifoldsThe main purpose of this paper is to (partially) generalize the two results to manifolds with local bounded Ricci covering geometry (Definition 0.2). Our two generalized results have been used in a recent work by Rong ([Ro4]).A fibration theorem in [Fu1] ([CFG]) says:Theorem 0.1. Given constants, n, i 0 > 0, there exist constant δ(n, i 0 ) > 0 (depending on n and i 0 ) and C(n) > 0, such that if a compact n-manifold M and a compact k-manifold N , k ≤ n, satisfywhere inj N denotes the injectivity radius of N and d GH denotes the Gromov-Hausdorff distance, then there is a smooth fiber bundle map, f : M → N , such that (0.1.1) f is a Ψ(δ|n, i 0 )-GHA (Gromov-Hausdorff approximation), where the function Ψ(δ|n, i 0 ) → 0 as δ → 0 while n and i 0 are fixed.Note that (0.1.1) and (0.1.3) imply that the induced metric on an f -fiber is almost flat, and thus each f -fiber is diffeomorphic to an infra-nilmanifold ([Gr], [Ru]). If one replaces N by a compact length space X and δ = δ(X), then there is a singular fibration map, f : M → X ([Fu2], [CFG]). For a r-distance ball at x ∈ M , B r (x), let π : ( B r (x),x) → (B r (x), x), denote the (incomplete) Riemannian universal cover. Put vol(B r (x)) vol(B r (x)), called the local rewinding volume of B r (x) ([Ro2]). Then vol(B r (x)) ≥ vol(B r (x)) and "=" if and only if B r (x) is simply connected. Definition 0.2. An n-manifold M is said to satisfy a local (r, v)-bound Ricci covering geometry, if there are constants, r, v > 0, such that Ric M ≥ −(n − 1), vol(B r (x)) ≥ v, ∀ x ∈ M.

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

“…This theorem summarizes the contributions from [48] and [77]. In [48], the existence of the topological fiber bundle map f is obtained using the canonical Reifenberg method in [18].…”

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

“…This theorem summarizes the contributions from [48] and [78]. In [48], the existence of the topological fiber bundle map f is obtained using the canonical Reifenberg method in [18].…”

“…This theorem summarizes the contributions from [48] and [78]. In [48], the existence of the topological fiber bundle map f is obtained using the canonical Reifenberg method in [18]. However, the infranil manifold structure of the fiber obtained in [78] is not a direct application of Theorem 2.7, since an f -fiber may have no uniform Ricci curvature lower bound.…”

“…Considerations on the ambient geometry is instead carried out in [78] -this work, when restricting to the case of almost flat manifolds, provides for the first time an approach entirely different from the original one in [39,79]; see also [77]. Notice that the arguments in [48,78] are local and our statements here are valid even if N is an open set in an k-dimensional manifold.…”

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

“…We expect our construction here to provide new perspectives on the Reifenberg method, which has far-reaching influences in geometric analysis and is still quickly evolving right now(cf. [21] and [60]).…”