Regularity of limits of noncollapsing sequences of manifolds

By V. Kapovitch. 2001

72 citations recorded by the source corpus.

Abstract. We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space Xn with $ n \ge 5 $ there exists an Alexandrov space Y homeomorphic to X which cannot be obtained as such a limit.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5922bb71feca8886bc1a34b8c30a59f04f3fef74

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In Applied Mathematics

1. Hui-Chun Zhang
2. Xiping Zhu
3. Anton Petrunin
4. V. Kapovitch
5. Koichi Nagano
6. F. Schulze
7. Miles Simon
8. A. Lytchak

In Geometry and Topology

1. Anton Petrunin
2. Hui-Chun Zhang
3. Xiping Zhu
4. A. Lytchak
5. Koichi Nagano
6. F. Schulze
7. Miles Simon
8. John Harvey

Citations in the corpus

• Geodesically complete spaces with an upper curvature bound (2018, background): “As a consequence, GCBA spaces which arise as limits of Riemannian manifolds can be very well understood, similarly to [Kap02].”
• Perelman???s Stability Theorem (2007, background): “We’ll need the following result from [Kap02] (cf.”
• Yau's gradient estimates on Alexandrov spaces (2010, background): “th φ′(t) = 1 for t 6 r0 − δ, φ′(t) = 1/2 for t > r0 +δand φ′′(t) = −1/(4δ) for t ∈ (r0 −δ,r0 +δ). The assertions (i) and (ii) have been proved for some positive constant r1 ≪ r0 in [34], (see also [15] for more details). The assertion (iii) is implicitly claimed in Petrunin’s manuscript [40]. Her”
• Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature (2012, background): “X¯. Let ¯h : X¯ → Rbe a function built up from distance functions such that it is strictly concave on some U containing p¯ and attains its maximum value on U at p¯ (see [40] for the construction, cf. [19, 24] for more explicit constructions). Let r0 be such that Br0(¯p) ⊂ ¯h−1 ([a,∞)) ⊂ U for some a”
• Semiconcave Functions in Alexandrov???s Geometry (2013, methodology): “Moreover, from [Kapovitch 2002] one has that any space of directions of a smoothable Alexandrov’s space is homeomorphic to the sphere.”
• Expanding solitons with non-negative curvature operator coming out of cones (2010, background): “1 of [22] says that Σ0(X) is homeomorphic to Σp0Mi (for i big enough) which is isometric to the standard sphere S since the (Mi, gi) are smooth manifolds.”