Restrictions on collapsing with a lower sectional curvature bound

Kapovitch, Vitali

doi:10.1007/s00209-004-0715-3

Cited by 15 publications

(12 citation statements)

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“…for some ε < r. In other words, we shall show that B M 3 α (x α , ε) looks like a fat solid torus with a shrinking core, i.e., In fact, using the conic lemma (Theorem 0.6 above), Kapovitch [22] already established a circle-fibration structure over the annular region A X 2 (x ∞ , δ, ε). Let x (X) denote the space of unit directions of an Alexandrov space X of curvature ≥ −1 at point x.…”

Section: Brief Introduction To Perelman's Mcs Theory and Applicationsmentioning

confidence: 96%

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

Cao

2010

J Geom Anal

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We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi (J. Differ. Geom. 56:1-66, 2000; Math. Ann. 333: 131-155, 2005) and Morgan-Tian (arXiv:0809.4040v1 [math.DG], 2008). A version of Perelman's collapsing theorem states: "Let {M 3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) andi is closed or has possibly convex incompressible toral boundary. Then M 3 i must be a graph manifold for sufficiently large i". This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman's critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.The verification of Perelman's collapsing theorem is the last step of Perelman's proof of Thurston's geometrization conjecture on the classification of 3-manifolds. A version of the geometrization conjecture asserts that any closed 3-manifold admits a piecewise locally homogeneous metric. Our proof of Perelman's collapsing theorem is accessible to advanced graduate students and non-experts.

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Section: Brief Introduction To Perelman's Mcs Theory and Applicationsmentioning

confidence: 96%

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

Cao

2010

J Geom Anal

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“…In the above argument, we employ the distance function from S( p, R) to prove Theorem 1.2. Similarly, one can use the averaged distance function constructed in [Perelman 1993] and [Kapovitch 2005] to prove Theorem 1.2.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Untitled

2014

Pacific J. Math.

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We demonstrate the relevance of the Prokhorov inequality to the study of Khintchine-type inequalities in symmetric function spaces. Our main result shows that the latter inequalities hold for a pair of quasi-Banach symmetric function spaces X and Y if and only if the Kruglov operator K acts from X to Y . We also obtain an extension of von Bahr-Esseen and Esseen-Janson L p -estimates for sums of independent mean zero random variables to the class of quasi-Banach symmetric spaces. In particular, in contrast to the well-known Esseen-Janson theorem, we do not assume that the summands are equidistributed.

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“…Note that the tangent cone at p splits: 20 Since gradient curves preserve extremal subsets q ∈ ∂A (see property 3.1.1 on page 18). Clearly |pq| = O(τ ), therefore applying the comparison from section 3.2 (or Corollary 3.1.3 if κ = 0 ) together with ( * ), we get…”

Section: Applicationsmentioning

confidence: 99%