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12
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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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.
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 7 (1 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
The canonical shrinking soliton associated to a Ricci flow
, 2008
"... 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 ..."
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Cited by 5 (1 self)
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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 [18], and McCann and the second author [12]; we briefly survey the link between these subjects.
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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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.
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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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.
OPTIMAL TRANSPORT AND CURVATURE
"... These notes record the six lectures for the CIME Summer Course held by the second author in Cetraro during the week of June 2328, 2008, with minor modifications. Their goal is to describe some recent developements in the theory of optimal transport, and their applications to differential geometry. ..."
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These notes record the six lectures for the CIME Summer Course held by the second author in Cetraro during the week of June 2328, 2008, with minor modifications. Their goal is to describe some recent developements in the theory of optimal transport, and their applications to differential geometry. We will focus on two main themes:
MASS TRANSPORT GENERATED BY A FLOW OF GAUSS MAPS
, 803
"... Abstract. Let A ⊂ R d be a compact convex set and let µ = 0dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = 1dx be another probability measure with supp(ν) = Br = {x: x  ≤ r}. There exists a mapping T such that ν = µ ◦ T −1 and T = ϕ · n, where ϕ: A → [ ..."
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Abstract. Let A ⊂ R d be a compact convex set and let µ = 0dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = 1dx be another probability measure with supp(ν) = Br = {x: x  ≤ r}. There exists a mapping T such that ν = µ ◦ T −1 and T = ϕ · n, where ϕ: A → [0, r] is a continuous potential with convex level sets and n is the Gauss map of the corresponding level sets of ϕ. We give a proof of this result based on the optimal transportation techniques. We show that in the case of smooth ϕ the level sets of ϕ are driven by the Gauss curvature flow ˙x(s) = −sd−1 1(sn) 0(x) K(x) · n(x). As a byproduct we reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.