24 citations · 59 across the 20 of their papers we have counts for
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Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature
Cuifang Si, Shicheng Xu
Let be a sequence of non-collapsed -manifolds with two-sidedly bounded Ricci curvature. We show that the Gromov-Haudorff limit space, , of the associated sequence of or…
Rigidity for Einstein manifolds under bounded covering geometry
Cuifang Si, Shicheng Xu
In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold must be flat if it is Einstein, i.e. $\opera…
Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology
Shicheng Xu
Based on a quantitative version of the classical Hopf-Rinow theorem in terms of the doubling property, we prove new precompactness principles in the (pointed) Gromov-Hausdorff topo…
Canonical nilpotent structure under bounded Ricci curvature and Reifenberg local covering geometry over regular limits
Zuohai Jiang, Lingling Kong, Shicheng Xu
It is known that a closed collapsed Riemannian -manifold of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of C…
Quantitative rigidity of almost maximal volume entropy for both RCD spaces and integral Ricci curvature bound
Lina Chen, Shicheng Xu
The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $…
Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry I
Zuohai Jiang, Lingling Kong, Shicheng Xu
We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the $C…
Total squared mean curvature of immersed submanifolds in a negatively curved space
Yanyan Niu, Shicheng Xu
Let and be two integers. Let be an isometrically immersed closed -submanifold of co-dimension that is homotopic to a point in a complete manifold ,…
1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold
Yingxiang Hu, Shicheng Xu
Let be a closed convex hypersurface lying in a convex ball of the ambient -manifold . We prove that, by pinching Heintze-Reilly's inequality via sect…
Recognizing shape via 1st eigenvalue, mean curvature and upper curvature bound
Yingxiang Hu, Shicheng Xu
Let be a closed immersed hypersurface lying in a contractible ball of the ambient -manifold . We prove that, by pinching Heintze-Reilly's inequality…
Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces
Shicheng Xu, Xuchao Yao
We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov -space with curvature bounded below, i.e., small loops at generate a subgroup of…
Collapsed Manifolds With Ricci Bounded Covering Geometry
Hongzhi Huang, Lingling Kong, Xiaochun Rong +1
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform…
A Geometric Approach to the Modified Milnor Problem
Lina Chen, Xiaochun Rong, Shicheng Xu
The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a pos…
Stability of pure Nilpotent Structures on collapsed Manifolds
Zuohai Jiang, Shicheng Xu
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 -manifol…
A parametrized compactness theorem under bounded Ricci curvature
Xiang Li, Shicheng Xu
We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.
Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold
Shicheng Xu
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifol…
Quantitative Volume Space From Rigidity with lower Ricci curvature bound II
Lina Chen, Xiaochun Rong, Shicheng Xu
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed -manifold of Ricci curvatur…
Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound
Lina Chen, Xiaochun Rong, Shicheng Xu
Let be a compact -manifold of ( is a constant). We are concerned with the following space form rigidity: is isometric to a space form…
An injectivity radius estimate in terms of metric sphere
Shicheng Xu
In this paper we prove that if a point in a complete Riemannian manifold is not a cut point of any point whose distance to is , then the injectivity radius of is str…
On conjugate points and geodesic loops in a complete Riemannian manifold
Shicheng Xu
A well-known Lemma in Riemannian geometry by Klingenberg says that if is a minimum point of the distance function to in the cut locus of , then eith…
Homotopy lifting property of an -Lipschitz and co-Lipschitz map
Shicheng Xu
An -Lipschitz and co-Lipschitz map, as a metric analogue of an -Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounde…