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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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Cited by 92 (0 self)
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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
Evolution of an extended Ricci flow system
, 2006
"... We show that Hamilton’s Ricci flow and the static Einstein vacuum equations are closely connected by the following system of geometric evolution equations: ∂tg = −2Rc(g) + 2αndu ⊗ du, ∂tu = Δgu, where g(t) is a Riemannian metric, u(t) a scalar function and αn a constant depending only on the dimens ..."
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Cited by 27 (0 self)
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We show that Hamilton’s Ricci flow and the static Einstein vacuum equations are closely connected by the following system of geometric evolution equations: ∂tg = −2Rc(g) + 2αndu ⊗ du, ∂tu = Δgu, where g(t) is a Riemannian metric, u(t) a scalar function and αn a constant depending only on the dimension n ≥ 3. This provides an interesting and useful link from problems in lowdimensional topology and geometry to physical questions in general relativity. 1.
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.
Ricci Flow with Surgery on Fourmanifolds with Positive Isotropic Curvature
, 2005
"... In this paper we study the Ricci flow on compact fourmanifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is twofold. One is to give a complete proof of the main theorem of Hamilton in [17]; the other is to extend some results of Perelman [26], [ ..."
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Cited by 26 (7 self)
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In this paper we study the Ricci flow on compact fourmanifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is twofold. One is to give a complete proof of the main theorem of Hamilton in [17]; the other is to extend some results of Perelman [26], [27] to fourmanifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper [27] on the Ricci flow. We also establish a uniqueness theorem for the Ricci flow on complete noncompact manifolds.
Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild . . .
"... The elliptic EinsteinDeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. RicciDeTurck flow is a constructive algorithm to solve this equation, and is simple to imp ..."
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Cited by 22 (5 self)
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The elliptic EinsteinDeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. RicciDeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the EinsteinDeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, KaluzaKlein, locally AdS or have extremal horizons. Using a maximum principle we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that RicciDeTurck flow preserves these classes of manifolds. As an example we simulate RicciDeTurck flow for a manifold with asymptotics relevant for AdS5/CFT4. Our maximum principle dictates there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N2c) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.
Ricci flow of negatively curved incomplete surfaces
, 2009
"... We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of wellposedness in this class. ..."
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Cited by 18 (6 self)
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We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of wellposedness in this class.
Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities
, 2010
"... We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting wellposedness issues that arise in the 2D case. On one hand we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or ..."
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Cited by 12 (2 self)
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We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting wellposedness issues that arise in the 2D case. On one hand we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or by contracting them. On the other hand, by adding a noncollapsedness assumption for the initial metric, we establish a uniqueness result.
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