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Strong uniqueness of the Ricci flow
 arXiv:0706.3081. HUAIDONG CAO
"... In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow onR 3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t)≡E. 1 ..."
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In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow onR 3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t)≡E. 1
Uniqueness of the Ricci flow on complete noncompact manifolds
, 2005
"... The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result t ..."
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Cited by 54 (5 self)
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The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.
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.
Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow
, 2006
"... Mean curvature flow evolves isometrically immersed base Riemannian manifolds M in the direction of their mean curvature in an ambient manifold ¯ M. We consider the classical solutions to the mean curvature flow. If the base manifold M is compact, the short time existence and uniqueness of the mean c ..."
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Cited by 10 (1 self)
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Mean curvature flow evolves isometrically immersed base Riemannian manifolds M in the direction of their mean curvature in an ambient manifold ¯ M. We consider the classical solutions to the mean curvature flow. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are wellknown. For complete noncompact isometrically immersed hypersurfaces M (uniformly local lipschitz) in Euclidean space, the short time existence was established by Ecker and Huisken in [9]. The short time existence and the uniqueness of the solutions to the mean curvature flow of complete isometrically immersed manifolds of arbitrary codimensions in the Euclidean space are still open questions. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. More precisely, let ( ¯ M, ¯g) be a complete Riemannian manifold of dimension ¯n such that the curvature and its covariant derivatives up to order 2 are bounded and the injectivity radius is bounded from below by a positive constant, we prove that the solution of the mean curvature flow with bounded second fundamental form on an isometrically immersed manifold M (may be high codimension) is unique. In the second part of the paper, inspired by the Ricci flow, we prove the pseudolocality theorem of mean curvature flow. As a consequence, we obtain the strong uniqueness theorem, which removes the boundedness assumption of the second fundamental form of the solution in the uniqueness theorem. 1
RICCI FLOW OF NONCOLLAPSED THREE MANIFOLDS WHOSE RICCI CURVATURE IS BOUNDED FROM BELOW
, 2009
"... In this paper we consider complete noncollapsed three dimensional Riemannian manifolds (M3, g) with Ricci curvature bounded from below and controlled growth: that is (M3, g) satisfies (a) Ricci(g) ≥ k for some k ∈ R (Ricci curvature bounded from below), (b) vol ( g B1(x)) ≥ v0> 0 for all x ∈ ..."
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In this paper we consider complete noncollapsed three dimensional Riemannian manifolds (M3, g) with Ricci curvature bounded from below and controlled growth: that is (M3, g) satisfies (a) Ricci(g) ≥ k for some k ∈ R (Ricci curvature bounded from below), (b) vol ( g B1(x)) ≥ v0> 0 for all x ∈ M, for some v0 ∈ R + (non collapsed), (c) supg Br(x) Riem(g)  ≤ h(r) (controlled growth) where h(r) = exp oexp o... exp(r) is the composition of the exponential function l times with itself for some l ∈ N. If (M3, g0) is in this class, then we show that there exist a constant T(v0)> 0 and a smooth solution to Ricci flow (M3, g(t)) t∈[0,T) with (M3, g(0)) = (M3, g0). The solution satisfies certain curvature decay estimates (Riem(g(t))  ≤ c for all t ∈ [0, T) ) and the t manifold remains noncollapsed ( vol ( g B1(x, t)) ≥ v0/2> 0 for all t ∈ [0, T)). This enables us to construct a Ricci flow of any (possibly singular) metric space (X, dX) which arises as the Gromov Hausdorff limit of a sequence of three dimensional Riemannian manifolds (M3 i, gi(0)) satisfying (a),(b) and (c). As a corollary we show that such an X must be a manifold.
Fourmanifolds with 1/4pinched flag curvatures
 Asian J. Math
, 2009
"... Abstract. The Ricci flow on a compact fourmanifold preserves the condition of pointwise 1/4pinching of flag curvatures. Any compact Riemannian fourmanifold with 1/4pinched flag curvatures is either isometric to CP2 or diffeomorphic to a spaceform. Key words. AMS subject classifications. 1. Intr ..."
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Cited by 7 (0 self)
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Abstract. The Ricci flow on a compact fourmanifold preserves the condition of pointwise 1/4pinching of flag curvatures. Any compact Riemannian fourmanifold with 1/4pinched flag curvatures is either isometric to CP2 or diffeomorphic to a spaceform. Key words. AMS subject classifications. 1. Introduction. It
ON 4DIMENSIONAL GRADIENT SHRINKING SOLITONS
, 2007
"... In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman ..."
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In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four.
SPHERE THEOREMS IN GEOMETRY
, 2009
"... In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques ..."
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Cited by 5 (1 self)
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In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton’s Ricci flow. We also obtain here new results concerning complete manifolds with pinched curvature.
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.