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77
THE KÄHLERRICCI FLOW AND THE ¯∂ OPERATOR ON VECTOR FIELDS
"... The limiting behavior of the normalized KählerRicci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is sh ..."
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Cited by 41 (12 self)
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The limiting behavior of the normalized KählerRicci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¯ ∂ † ¯ ∂ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C∞ to a KählerEinstein metric.
RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
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Cited by 37 (1 self)
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Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be nonincreasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.
RECENT PROGRESS ON RICCI SOLITONS
, 2009
"... In recent years, there has seen much interest and increased research activities in Ricci solitons. Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton’s Ricci flow and play important roles in the singularity study of the Ricci flow. In this pap ..."
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Cited by 30 (0 self)
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In recent years, there has seen much interest and increased research activities in Ricci solitons. Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton’s Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.
Sharp logarithmic Sobolev inequalities on gradient solitons and applications
, 2008
"... We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by ..."
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Cited by 27 (0 self)
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We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. 1
Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics
, 2007
"... By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the twodimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomp ..."
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Cited by 23 (7 self)
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By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the twodimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert for wellposedness is that the flow should be complete for all positive times; our discussion of uniqueness also invokes pseudolocality.
A GRADIENT ESTIMATE FOR ALL POSITIVE SOLUTIONS OF THE CONJUGATE HEAT EQUATION UNDER RICCI FLOW
, 2006
"... Abstract. We establish a pointwise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman’s pointwise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman’s estimate, the most general form of our ..."
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Cited by 22 (0 self)
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Abstract. We establish a pointwise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman’s pointwise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman’s estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the LiYau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A generalization of Perelman’s Wentropy is defined in both the Ricci flow and fixed metric case. We also find a new family of heat kernel estimates. Contents
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.