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Dimensional reduction and the long-time behavior of Ricci flow
- COMM. MATH. HELV
, 2007
"... If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that are O(t−1) and diameter that is O(t 1 2), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton. ..."
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Cited by 18 (4 self)
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If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that are O(t−1) and diameter that is O(t 1 2), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.
A SIMPLE PROOF OF PERELMAN’S COLLAPSING THEOREM FOR 3-MANIFOLDS
"... Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. 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 ..."
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Cited by 8 (2 self)
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Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. 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) and diam(M 3 i) ≥ c0> 0. Suppose that all unit metric balls in M 3 i have very small volume at most is closed or it vi → 0 as i → ∞ and suppose that either M 3 i 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 implicit function theorem. Among other things, we use Perelman’s critical point theory (e.g., multiple conic singularity theory and his fibration theory) for 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. Our proof of Perelman’s collapsing theorem is accessible to non-experts and advanced graduate students. Contents
LONG-TIME BEHAVIOR OF 3 DIMENSIONAL RICCI FLOW D: PROOF OF THE MAIN RESULTS
"... Abstract. This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and ..."
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Cited by 1 (0 self)
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Abstract. This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by Ct−1. The second result provides a qualitative description of the geometry as t→∞. Contents
A PROOF OF PERELMAN’S COLLAPSING THEOREM FOR 3-MANIFOLDS
, 2009
"... We will simplify the earlier proofs of Perelman’s collapsing theorem of 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states that: “Let {M 3 i} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below ..."
Abstract
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We will simplify the earlier proofs of Perelman’s collapsing theorem of 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states that: “Let {M 3 i} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M 3 i) ≥ c0> 0. Suppose that all unit metric balls in M 3 i have very small volume at most vi → 0 as i → ∞ and suppose that either M 3 i is closed or it has possibly convex incompressible tori boundary. Then M 3 i must be a graph-manifold for sufficiently large i”. This result can be viewed as an extension of implicit function theorem. Among other things, we use Perelman’s semi-convex analysis of distance functions 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-maniflds. Our proof of Perelman’s collapsing theorem is almost self-contained. We believe that our proof of this collapsing theorem is accessible to non-experts and advanced graduate students.
