Nilpotent structures and invariant metrics on collapsed manifolds
TL;DR: In this article, a complete Riemannian manifold of bounded curvature, called Mn, was considered, and the complementary set was defined as the e-collapsed part of the manifold.
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Abstract: Let Mn be a complete Riemannian manifold of bounded curvature, say IKI 0, we put Mn = W n(c) U Fn (c), where ?1n (e) consists of those points at which the injectivity radius of the exponential map is > e The complementary set, &"n (c) is called the e-collapsed part of Mn If x E 4 (n), r < E, then the metric ball Bx (r) is quasi-isometric, with small distortion, to the flat ball B0(r) in the Euclidean space, Rn After slightly
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Citations
Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry I
Zuohai Jiang,Lingling Kong,Shicheng Xu +2 more
- 01 May 2022
TL;DR: Cheeger-Gromov's and Anderson's convergence theorem for regular limit spaces of collapsed manifolds with bounded Ricci curvature and local covering geometry was extended in this article . But the convergence theorem was not extended to regular limit space of collapsed manifold with local cover geometry.
1
A generalized π 2 -diffeomorphism finiteness theorem
Xiaochun Rong,Xuchao Yao +1 more
TL;DR: In this paper, Fang-X. Rong and A. Petrunin-W. Tuschmann generalized the π2-diffeomorphism finiteness result by removing the condition that π1(M) = 0 and asserting the diffeomorphism Finiteness on the Riemannian universal cover of M.
1
Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface
Lorenzo Foscolo
- 01 Jan 2020
TL;DR: In this article, the authors survey three constructions of degenerating sequences of hyperkahler metrics on the (smooth 4-manifold underlying a complex) K3 surface, the classical Kummer construction, Gross-Wilson's work on collapse along the fibres of an elliptic fibration, and the author's construction of sequences collapsing to a 3-dimensional limit.
1
On three-manifolds with bounded geometry
Michel Boileau,Daryl Cooper +1 more
- 01 Jan 2004
TL;DR: In this article, the authors combine some of Cheeger-Gromov's results from the geometry of collapses of Riemannian 3-manifolds together with some three-dimensional topology to prove results which say that there are at most finitely many diffeomorphism classes of prime nongeometrizable 3-Manifolds which admit a metric of bounded geometry (i.e. with bounded sectional curvatures and bounded volume).
1
Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates
TL;DR: In this paper, the Ricci flow is shown to collapse in the Gromov-Hausdorff sense to a space which is not a manifold but only a metric space.
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M. S. Raghunathan
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Deforming the metric on complete Riemannian manifolds
TL;DR: Soit (M,g ij (x)) une variete de Riemann a n dimensions complete non compacte de tenseur de complexe riemannien {R ijkl } satisfaisant: |R Ijkl | 2 ≤k 0 sur M, ou 0 0 dependant seulement de n, m and k 0 telle que l'equation d'evolution δg Ij (ex,t)|δt=−2R iJ (x,t), δt =−2
Finiteness theorems for riemannian manifolds.
TL;DR: In this article, it was shown that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism classes of compact n-dimensional manifolds admitting a metric for which these bounds are satisfied is finite.
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