Collapsed Manifolds With Ricci Bounded Covering Geometry

Huang, Hongzhi; Kong, Lingling; Rong, Xiaochun; Xu, Shicheng

doi:10.48550/arxiv.1808.03774

Cited by 7 publications

(3 citation statements)

References 9 publications

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“…We notice that distance distortion estimates similar to Proposition 5.3 and Theorem 5.4 were already obtained in Chen-Rong-Xu [23] and Huang-Kong-Rong-Xu [61]. where Rc min (x) is the minimum eigenvalue of Rc(x), T is a positive number, ǫ < ǭ(m) is a sufficiently small positive number.…”

Section: Distance Distortion and Topological Stabilitysupporting

confidence: 64%

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

Wang

2018

Cambridge Journal of Mathematics

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Section: Distance Distortion and Topological Stabilitysupporting

confidence: 64%

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

Wang

2018

Cambridge Journal of Mathematics

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“…For reader's convenience we present a direct short proof not relying on Theorem 7.1. It uses a somewhat different argument for the crucial upper diameter bound than that in [HKRX,Lemma 1.11] which in turn relied on [LC, Lemma 2.10].…”

Section: Smoothing Via Ricci Flowmentioning

confidence: 99%

Mixed curvature almost flat manifolds

Kapovitch

2019

Preprint

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“…When (1.1) is weakened to a mere Ricci curvature lower bound, while this strategy works in certain situations like controlling the fundamental group or understanding the infinitesimal behavior of the metric (see e.g. [26] and [23]), it cannot lead us to the more delicate gradient estimate as (1.3), without any extra assumption (like Ricci bounded covering geometry). This is mainly due to the lack of a fibration (or submersion) structure, as well as the absence of the structural information of the Φ fibers.…”

Section: Introduction and Backgrounddmentioning

confidence: 99%

Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

Huang,

Taskent

2019

Preprint

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By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting map, in the L 2 -average sense. This generalizes an estimate of Fukaya in the case of collapsing with bounded diameter and sectional curvature.Here we recall that when the Riemannian manifold (M i , g) is sufficiently close to, in the Gromov-Hausdorff sense, another lower diemnsional Riemannian manifold (N, h), then the regularity assumptions (1.1) guarantee that there is a fibration Φ i : M i → N, which is also an almost Riemannian submersion. Moreover, the Φ i fibers, as embedded submanifolds in M i , are all homeomorphic

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Collapsed Manifolds With Ricci Bounded Covering Geometry

Cited by 7 publications

References 9 publications

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

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

Mixed curvature almost flat manifolds

Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

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