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49
Width and finite extinction time of Ricci flow
, 2007
"... This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M i ..."
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Cited by 26 (1 self)
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This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M is a homotopy 3sphere, the width is loosely speaking the
Recent Developments on Hamilton’s Ricci flow
 SURVEYS IN DIFFERENTIAL GEOMETRY XII
, 2008
"... In 1982, Hamilton [41] introduced the Ricci flow to study compact threemanifolds with positive Ricci curvature. Through decades of works of many mathematicians, the Ricci flow has been widely used to study the topology, geometry and complex structure of manifolds. In particular, Hamilton’s fundamen ..."
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Cited by 26 (6 self)
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In 1982, Hamilton [41] introduced the Ricci flow to study compact threemanifolds with positive Ricci curvature. Through decades of works of many mathematicians, the Ricci flow has been widely used to study the topology, geometry and complex structure of manifolds. In particular, Hamilton’s fundamental works (cf. [12]) in the past two decades and the recent breakthroughs of Perelman [80, 81, 82] have made the Ricci flow one of the most intricate and powerful tools in geometric analysis, and led to the resolutions of the famous Poincare ́ conjecture and Thurston’s geometrization conjecture in threedimensional topology. In this survey, we will review the recent developments on the Ricci flow and give an outline of the HamiltonPerelman proof of the Poincare conjecture, as well as that of a proof of Thurston’s geometrization conjecture.
HamiltonPerelman’s Proof of the Poincaré Conjecture and The Geometrization Conjecture
, 2006
"... In this paper, we provide an essentially selfcontained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of threemanifolds. In particular, we give a detailed exposition of a complete pro ..."
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Cited by 20 (0 self)
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In this paper, we provide an essentially selfcontained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of threemanifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.
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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Cited by 18 (4 self)
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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.
An excursion into geometric analysis
 SURVEYS IN DIFFERENTIAL GEOMETRY
, 2003
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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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Cited by 16 (4 self)
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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.
Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3manifolds
, 2008
"... We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3manifolds. Our main result is that such actions on elliptic and hyperbolic 3manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott ..."
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Cited by 15 (0 self)
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We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3manifolds. Our main result is that such actions on elliptic and hyperbolic 3manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [MS86], it follows that such actions on geometric 3manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [Th82].
AREAMINIMIZING PROJECTIVE PLANES IN THREEMANIFOLDS
, 909
"... Let M be a compact threemanifold equipped with a Riemannian metric g. We denote by F the set of all embedded incompressible projective planes in M. In other words, F consists of all embedded surfaces Σ ⊂ M such that Σ is homeomorphic to RP 2 and the induced map i # : π1(Σ) → π1(M) is ..."
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Cited by 12 (6 self)
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Let M be a compact threemanifold equipped with a Riemannian metric g. We denote by F the set of all embedded incompressible projective planes in M. In other words, F consists of all embedded surfaces Σ ⊂ M such that Σ is homeomorphic to RP 2 and the induced map i # : π1(Σ) → π1(M) is