John Lott
We show that a certain entropy-like 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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An $ε$-regularity theorem for Perelman's reduced volume
本文研究Ricci流中Perelman约化体积的正则性，证明当其接近1时基点曲率半径具有下界。
In recent years, there has seen much interest and increased research activities on Perelman's paper. Section one and two of this paper aim to establish Perelman's local non-collapsing result for the Ricci flow. This will provide a positive lower bound on the injectivity radius for the Ricci flow under blow-up analysis. We also discuss the gradient flow formalism of the Ricci flow and Perelman's motivation from physics. In this paper, we survey some of the recent progress on Perelman's functional, reduced volume and reduced length.
The main purpose of this paper is to present a number of analytic and geometric properties of the $l$-function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the reduced volume.
In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\to 0$ as $d(x)=d_g(x,p)\to \infty$, then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.
In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow.
In this paper, we show that any 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 Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained.
Deriving Perelman's entropy from Colding's monotonic volume
本文通过将 Colding 的单调体积适当应用于 Perelman 的 $N$-空间，证明了 Perelman 的熵可表示为 Ricci 平坦流形上调和函数的 Colding 单调体积的极限，从而揭示了熵的起源。
In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a Kähler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the Kähler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).
In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate
Perelman has discovered two integral quantities, the shrinker entropy $\cW$ and the (backward) reduced volume, that are monotone under the Ricci flow $\pa g_{ij}/\pa t=-2R_{ij}$ and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entropy} $\ctW$ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals $μ_+$ and $ν_+$. The {\it forward reduced volume}…
The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volume doubling property are all consequences of an entropy inequality recently discovered by Baudoin-Garofalo. The latter is a linearized version of a logarithmic Sobolev inequality that is due to D. Bakry and M. Ledoux.
Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $δ_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $α_M := \lim_{r \to \infty} \frac{\Vol(B_p(r))}{ω_n r^n} \geq 1-δ_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $α(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.
In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This establishes a relation between these two monotone quantities.
In this note, we describe a new link between Perelman's monotonicity formula for the reduced volume and ideas from optimal transport theory.
Linear stability of Perelman's $ν$-entropy of standard Einstein manifolds
本文针对标准爱因斯坦流形，估计拉普拉斯-贝尔特拉米算子最小正特征值，证明Schwahn发现的112个稳定爱因斯坦流形中除7个外均满足条件，故对佩雷尔曼ν-熵线性稳定。
In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4π)^{\frac{n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.
Zamolodchikov's c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be gradient one to all loop orders. However, the monotonicity of the flow of the target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman (math.dg/0211159) by constructing an ``entropy'' functional which is essentially the metric-dilaton action extremised with respect to the dilaton with a cond…
We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results on volume non-collapsing.
On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces
本文将Perelman的W-熵公式、Shannon熵幂凹性及Li-Yau-Hamilton-Perelman Harnack不等式推广到度量测度空间上的超Ricci流，并证明RCD(0,N)空间中体积非局部坍缩与W-熵下界等价，进而研究最优对数Sobolev不等式。
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-Hölder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.
Suppose $M$ is a complete n-dimensional manifold, $n\ge 2$, with a metric $\bar{g}_{ij}(x,t)$ that evolves by the Ricci flow $\partial_t \bar{g}_{ij}=-2\bar{R}_{ij}$ in $M\times (0,T)$. For any $0<p<1$, $(p_0,t_0)\in M\times (0,T)$, $q\in M$, we define the $\Cal{L}_p$-length between $p_0$ and $q$, $\Cal{L}_p$-geodesic, the generalized reduced distance $l_p$ and the generalized reduced volume $\widetilde{V}_p(τ)$, $τ=t_0-t$, corresponding to the $\Cal{L}_p$-geodesic at the point $p_0$ at time $t_0$. Under the condition $\bar{R}_{ij}\ge -c_1\bar{g}_{ij}$ on $M\times (0,t_0)$ for some constant $c…
In this note we will adapt Topping's $\mathcal{L}$-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold $(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M$. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List's (and hence also of Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.
We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motiv…
We prove a so called $κ$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $κ$ non-collapsing property for Ricci flow. These two results together imply volume doubling property for Ricci flow without assuming Ricci curvature lower bound.
We prove a volume inequality for 3-manifolds having C^0 metrics "bent" along a hypersurface, and satisfying certain curvature pinching conditions. The result makes use of Perelman's work on Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume orientable hyperbolic 3-manifold. An appendix by Dunfield compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed emb…
On a parabolic curvature lower bound generalizing Ricci flows
针对超Ricci流无法保留Perelman泛函单调性的问题，本文用最优传输刻画了Buzano提出的推广条件，证明其等价于多种分析不等式，且多数证明无需张量分析，为奇异情形推广提供可能。
$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these two. Due to the surprising similarity between those concepts in the Ricci flow and the linear heat equation, a natural question whether such equation holds for the Ricci flow ancient solution was asked by L. Ni. In this paper, we gave an alternative proof to L. Ni's original equat…
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a rescaling of the curvature term in the shrinker entropy. The second is associated with a gradient flow obtained by adding a curvature-drift to Perelman's backward heat equation. We show that the resulting Fokker-Planck PDE is the natural diffusion flow for probability measures absolu…
The Volume-Renormalized Mass from a Hamiltonian Perspective
本研究从约化哈密顿视角推导渐近双曲流形的体积重整化质量，将其定义拓展至初值数据集，约化ADM哈密顿量后还原出该质量，并证明其演化非递增、仅自相似时空下为常数。
In this paper, we define a reduced distance function based at a point at the singular time $T<\infty$ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition.
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