A parametrized compactness theorem under bounded Ricci curvature

Li, Xiang; Xu, Shicheng

doi:10.1007/s11464-017-0676-6

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Outline Of Proof Theorem 011

Notations and Preliminaries1

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2019

2023

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Cited by 2 publications

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“…Compared to the earlier stability results ([7, section 7], [23], [36], [25], etc. ), our main improvement here is that no fiberwise closeness (nor 1 -closeness) of are required.…”

Section: Outline Of Proof Theorem 01mentioning

confidence: 58%

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Stability of Pure Nilpotent Structures on Collapsed Manifolds

Jiang

2019

International Mathematics Research Notices

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The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a -manifold of bounded sectional curvature are 0 -bi-Lipchitz equivalent and sufficient collapsed (depending on 0 and ), then up to a diffeomorphism, the underlying nilpotent Killing structures coincide with each other or one is embedded into another as a subsheaf. It improves Cheeger-Fukaya-Gromov's locally compatibility of pure nilpotent Killing structures for one collapsed metric of bounded sectional curvature to two Lipschitz equivalent metrics. As an application, we prove that those pure nilpotent Killing structures constructed by various smoothing method to a Lipschitz equivalent metric of bounded sectional curvature are uniquely determined by the original metric modulo a diffeomorphism.

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“…Compared to the earlier stability results ([7, section 7], [23], [36], [25], etc. ), our main improvement here is that no fiberwise closeness (nor 1 -closeness) of are required.…”

Section: Outline Of Proof Theorem 01mentioning

confidence: 58%

“…By standard variation methods (cf. Lemma 1 in [25]), d 2 is under control by , , Lipschitz and co-Lipschitz constant of , and the sectional curvature bound of the base space , as follows.…”

Section: Notations and Preliminariesmentioning

confidence: 99%