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37
Optimal transport and Perelman’s reduced volume
, 2008
"... We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume. ..."
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Cited by 30 (2 self)
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We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
Perelman’s reduced volume and gap theorem for Ricci flow
"... Abstract. In this paper, we show that an ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of M. Anderson’s result for Ricci flat manifolds ..."
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Cited by 16 (1 self)
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Abstract. In this paper, we show that an ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of M. Anderson’s result for Ricci flat manifolds. As a corollary, a gap theorem for shrinking gradient Ricci solitons is obtained. 1.
The Canonical Expanding Soliton and Harnack inequalities for Ricci flow
, 2009
"... We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle. 1 ..."
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Cited by 11 (2 self)
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We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle. 1
Horizontal diffusion in C1 path space
"... Summary. We define horizontal diffusion in C1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction pr ..."
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Cited by 9 (0 self)
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Summary. We define horizontal diffusion in C1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the MongeKantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise. Key words: Brownian motion, damped parallel transport, horizontal diffusion,
The canonical shrinking soliton associated to a Ricci flow
 Calc. Var
"... Abstract To every Ricci flow on a manifold M over a time interval I ⊂ R−, we associate a shrinking Ricci soliton on the spacetime M×I. We relate properties of the original Ricci flow to properties of the new higherdimensional Ricci flow equipped with its own timeparameter. This geometric constru ..."
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Cited by 8 (2 self)
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Abstract To every Ricci flow on a manifold M over a time interval I ⊂ R−, we associate a shrinking Ricci soliton on the spacetime M×I. We relate properties of the original Ricci flow to properties of the new higherdimensional Ricci flow equipped with its own timeparameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author
Horizontal diffusion in C 1 path space
 Probabilités, Lecture Notes in Mathematics
"... Abstract. We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction ..."
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Cited by 4 (0 self)
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Abstract. We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the MongeKantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.
New logarithmic Sobolev inequalities and an εregularity theorem for the Ricci flow (2012), available at arXiv:1205.0380
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