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37
Perelman’s stability theorem
, 2007
"... Abstract. We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv � k GromovHausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i. 1. ..."
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Abstract. We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv � k GromovHausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i. 1.
Dimensional reduction and the longtime behavior of Ricci flow
 COMM. MATH. HELV
, 2007
"... If g(t) is a threedimensional 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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If g(t) is a threedimensional 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.
Linear stability of homogeneous Ricci solitons
 Int. Math. Res. Not. (2006), Art. ID 96253
"... Abstract. As a step toward understanding the analytic behavior of TypeIII Ricci ‡ow singularities, i.e. immortal solutions that exhibit j Rm j C=t curvature decay, we examine the linearization of an equivalent ‡ow at …xed points discovered recently by Baird–Danielo and Lott: nongradient homogeneous ..."
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Abstract. As a step toward understanding the analytic behavior of TypeIII Ricci ‡ow singularities, i.e. immortal solutions that exhibit j Rm j C=t curvature decay, we examine the linearization of an equivalent ‡ow at …xed points discovered recently by Baird–Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the ‡ow at these …xed points and prove that the linearizations generate C0 semigroups. 1.
Weak collapsing and geometrisation of aspherical 3manifolds
, 2008
"... Let M be a closed, orientable, irreducible, nonsimply connected 3manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred ..."
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Let M be a closed, orientable, irreducible, nonsimply connected 3manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman’s proof of the Geometrisation Conjecture for aspherical 3manifolds.
Total Curvatures of Model Surfaces Control Topology of Complete Open Manifolds with Radial Curvature Bounded Below. I
, 2009
"... We investigate the finiteness structure of a complete open Riemannian nmanifold M whose radial curvature at a base point of M is bounded from below by that of a noncompact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann f ..."
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We investigate the finiteness structure of a complete open Riemannian nmanifold M whose radial curvature at a base point of M is bounded from below by that of a noncompact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact set contains all critical points for any Busemann function on M. As corollaries by the main theorem, M has finite topological type, and the isometry group of M is compact.
Completion of the proof of the geometrization conjecture
"... This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a ..."
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This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a locally homogeneous Riemannian metric of finite volume. Recall that a Riemannian manifold is homogeneous if its isometry group acts transitively on the underlying manifold; a locally homogeneous Riemannian manifold is the quotient of a homogeneous Riemannian manifold by a discrete group of isometries acting freely. Recall also that a prime 3manifold is one which is not diffeomorphic to S 3 and which is not a connected sum of two manifolds neither of which is diffeomorphic to S 3. It is a classic result in 3manifold topology, see [12] that every 3manifold is a connected sum of a finite number of prime 3manifolds, and this decomposition is unique up to the order of the factors. The main part of this paper is devoted to giving a proof of Theorem 7.4 stated
Geometrisation of 3Manifolds
 In preparation
"... The aim of this book is to give a proof of Thurston’s Geometrisation Conjecture, solved by G. Perelman in 2003. Perelman’s work completes a program initiated by R. Hamilton, using a geometric evolution equation called Ricci flow. Perelman presented his ideas in three very concise manuscripts [Per02] ..."
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The aim of this book is to give a proof of Thurston’s Geometrisation Conjecture, solved by G. Perelman in 2003. Perelman’s work completes a program initiated by R. Hamilton, using a geometric evolution equation called Ricci flow. Perelman presented his ideas in three very concise manuscripts [Per02], [Per03a], [Per03b]. Important work has since been done to fill in the details. The first set of notes on Perelman’s papers was posted on the web in June 2003 by B. Kleiner and J. Lott. These notes have progressively grown to the point where they cover the two papers [Per02], [Per03b]. The final version has been published as [KL08]. A proof of the Poincaré Conjecture, following G. Perelman, is given in the book [MT07] by J. Morgan and G. Tian. Another text covering the Geometrisation Conjecture following Perelman’s ideas is the article [CZ06a] by H.D. Cao and X.P. Zhu. Alternative approaches to some of Perelman’s arguments were given by T. Colding and W. Minicozzi [CM07], T. Shioya and T. Yamaguchi [SY05], the
A SIMPLE PROOF OF PERELMAN’S COLLAPSING THEOREM FOR 3MANIFOLDS
"... Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds with curvature bounded from below ..."
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Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds 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 graphmanifold 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 3manifolds. The verification of Perelman’s collapsing theorem is the last step of Perelman’s proof of Thurston’s Geometrization Conjecture on the classification of 3manifolds. Our proof of Perelman’s collapsing theorem is accessible to nonexperts and advanced graduate students. Contents
Perelman’s proof of the Poincaré conjecture: a nonlinear PDE perspective
, 2006
"... We discuss some of the key ideas of Perelman’s proof of Poincaré’s conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations. ..."
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We discuss some of the key ideas of Perelman’s proof of Poincaré’s conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.