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A Flow Tangent to the Ricci Flow via Heat Kernels and Mass Transport
Functional Analysis · Submitted 29 Aug 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow.
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The abstract, sentence by sentence
We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. | contribution 0.96 | 0.91 | 0.61 | asserted 1.00 |
| s1 | We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below. | contribution 0.90 | 0.94 | 0.29 | asserted 1.00 |
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The paper
Original abstract
We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below.
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How the literature uses it
2 arXiv papers cite this in 23 sentences
Papers citing it · 15 sentences without a confident label
Builds on it 1 paper
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Let us observe that from ([ref]) it easily follows that ([ref]) can be rewritten as [equation] or equivalently as [equation] Also note that by differentiating both members of ([ref]) with respect to $t$ we to get, for any $t>0$, (cf. [ref] Prop. 4.1 for a similar computation for the standard heat kernel), [equation] Since $∇^(z) ∂∂tp^(ω) _t(y , z)= ∇^(z) △_ω^(y) p^(ω) _t(y , z)$, and the weighted laplacian $△_ω^(y)$ acts with respect to the variable point $y$, we can write, after an obvious rearrangement of terms, [equation] By inserting the defining pde ([ref]) in $△_ω^(y) [U_⊥(z)· ∇^(z)p^(ω) _t(y , z) ]$, we eventually get [equation] where we dropped the superscript $(y)$ since the operators $∇$, $div_ω$, and $△_ω$ all act with respect to $y$.
The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow · Heat kernel induced Ricci flow · 0.99
Uses its method or tool 1 paper
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Writing, as usual, $ψ_(t):=ψ_(t, c_λ, c'_λ)$ for ease of notation we have, by an obvious transposition of the argument provided in [ref] , Prop. 3.7, [equation] where, in the last line, we exploited the pde ([ref]) defining the potential $ψ_(t, c_λ, c'_λ)$ corresponding to the tangent vector $c'_λ$.
The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow · The induced G-M metric rescaling $(M,g)↦(M, g_t^(ω))$ · 0.86
What it cites · 5 sentences without a confident label
Cites as support 2 papers
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Indeed, a lower Ricci curvature bound seems necessary due to the fact that the heat flow is well defined and nicely behaves in relation with the $W_2$–geometry only in presence of the $CD(K,∞)$ condition (see [ref] ).
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below · The Construction in a Non–Smooth Setting · 0.99
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This choice rules rules out Finsler–type geometries and ensures, on one hand, the existence of a heat kernel, on the other hand, the $W_2$–contraction along two heat flows (in [ref] Sturm and Ohta proved that on a normed space $(^d,·,ℒ^d)$ the heat flow never contracts the $W_2$–distance unless the norm comes from a scalar product, therefore, due to the spirit of the discussion here, it is natural to avoid considering this sort of spaces).
Non-contraction of heat flow on Minkowski spaces · Introduction · 0.97
Cites as background 1 paper
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The class of spaces with linear heat flow has been investigated in [ref] and [ref] , the definition being the following.
Metric measure spaces with Riemannian Ricci curvature bounded from below · $RCD(K,∞)$ spaces · 0.86
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