Simple proofs of some results of Perelman on Ricci flow

Brownian motion on Perelman's almost Ricci-flat manifold

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold $\mathscr M=M\times \mathbb S^N\times I$, whose dimension depends on a parameter $N$ unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for $N \to \infty$ converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on $\mathscr M$ and for the horizontal Laplacian on the orthonormal frame bundle $\mathscr{OM}$ . As …

加权黎曼流形上沿超Perelman Ricci流的一类广义非线性热型方程的梯度估计

Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

本文研究沿$(k,m)$超Perelman-Ricci流演化的加权黎曼流形上一类广义非线性热型方程正解的梯度估计，推导了对应梯度估计式，并应用其得到Harnack型不等式与Liouville型定理。

Perelman's entropy on ancient Ricci flows

In [ZY2], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth

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.

A second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow

We show a quite simple second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow over Fano manifolds.

$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons

In this paper, we prove the characterization of the $(K, \infty)$-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dimension free Harnack inequality on manifolds with $(K, \infty)$-super Perelman Ricci flows. Based on a new second order differential inequality on the Boltzmann-Shannon entropy for the heat equation of the Witten Laplacian, we introduce a new $W$-entropy quantity and prove its mon…

Perelman标量曲率与Fano流形上Kähler-Ricci流直径估计的一个新证明

A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

本文为Fano流形上Kähler-Ricci流的Perelman标量曲率和直径估计提供新证明，基于时空函数族的新Harnack估计，并推广至一般有限时间解。

Singularity models in the three-dimensional Ricci flow

The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension $3$. More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension $3$ is modeled on an ancient $κ$-solution. Moreover, Perelman proved a structure theorem for ancient $κ$-solutions in dimension $3$. In this survey, we discuss recent developments which have led to a complete classification of all…

Perelman's reduced volume and a gap theorem for 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.

Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $λ(\cdot)$ for Kähler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $λ(\cdot)$ in Kähler class $2πc_1(M)$ under an assumption that the modified Mabuchi's K-energy $μ(\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow in [TZhu3].

Perelman-type no breather theorem for noncompact Ricci flows

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.

Perelman's functional and reduced volume

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.

Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective

We discuss some of the key ideas of Perelman's proof of Poincaré's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

Some gradient estimates for the heat equation on domains and for an equation by Perelman

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

Perelman's Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds

In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + infinity whenever the Yamabe invariant is positive.

Notes on Perelman's papers

These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".

从 Colding 的单调体积推导 Perelman 的熵

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。

Perelman's Entropy Functional at Type I Singularities of the Ricci Flow

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit W-density at a Type I singular point to the entropy of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.

关于推广Ricci流的一类抛物型曲率下界

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。

The Soliton-Ricci Flow over Compact Manifolds

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's $\mathcal{W}$-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's $\mathcal{W}$-functional over such subspaces…

Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend X. Cao's methods of eigenvalues\cite{C} and impro…

关于Perelman的Ricci收缩子熵的第二变分

On second variation of Perelman's Ricci shrinker entropy

本文给出Perelman的ν-熵第二变分公式的详细证明，纠正了已有文献中稳定性算子的错误，并通过相关Lichnerowicz型算子的最小特征值及其重数得到线性稳定收缩子的必要条件。

关于度量测度空间上超Ricci流的Perelman W-熵与Shannon熵幂

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。

Ricci flow on Poincare' and Thurston's Geometrization Conjecture

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

The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow

Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci flow. By exploiting the heat kernel embedding introduced by N. Gigli and C. Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between non-linear sigma m…

Perelman's entropy for some families of canonical metrics

We numerically calculate Perelman's entropy for a variety of canonical metrics on $\mathbb{CP}^{1}$-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.

Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are…

Uniqueness and stability of Ricci flow through singularities

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- i…

Pseudolocality for the Ricci flow and applications

In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compa…

From the Boltzmann $H$-theorem to Perelman's $W$-entropy formula for the Ricci flow

In 1870s, L. Boltzmann proved the famous $H$-theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of $W$-entropy and proved the $W$-entropy formula for the Ricci flow. This plays a crucial role in the proof of the no local collapsing theorem and in the final resolution of the Poincaré conjecture and Thurston's geometrization conjecture. In our previous paper \cite{Li11a}, the author gave a probabilistic interpretation of the $W$-entropy using the Boltzmann-Shannon-Nash en…

Perelman粘合定理的一个推广及其应用

A generalization of the Perelman gluing theorem and applications

本文将Perelman正Ricci曲率粘合定理推广到从正标量曲率到正截面曲率的正中间曲率条件，并应用于证明观察者模空间具有非平凡高阶同伦群及给出带全测地边界的正中间Ricci曲率度量存在性充分条件。

Perelman's entropy and doubling property on Riemannian manifolds

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.

On Perelman's Dilaton

By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the ``real'' part of the metric under a constrained gradient flow of the Einstein-Hilbert gravitational action in higher dimension.

Rotational symmetry of self-similar solutions to the Ricci flow

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

Perelman的$\u0026mathcal{W}$泛函在Kähler势空间上的凹性

Concavity of Perelman's $\mathcal{W}$-functional over the space of Kähler potentials

该研究指出Perelman的$\u0026mathcal{W}$泛函在Kähler势空间中Kähler-Ricci孤立子邻域的凹性可由Kähler-Ricci孤立子变分稳定性的已有结论直接推出，并利用前期工作的初等公式给出了简洁证明。

Minimally invasive surgery for Ricci flow singularities

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S^(n+1), without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities".

Some elementary consequences of Perelman's canonical neighborhood theorem

In this purely expository note, we recall a few known properties of 3-dimensional singularity models. These properties are direct consequences of Perelman's canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional kappa-solutions.

Negativity of Perelman's Li-Yau-Hamilton type expression

Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space.

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows.

Perelman约化体积的一个$\epsilon$-正则性定理

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。

A note on Perelman's no shrinking breather theorem

As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's $\mathcal{L}$-geometry. Our proof follows the argument in Lu and Zheng of constructing an ancient solution, and removes a technical assumption made by them.

A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

We establish a point-wise gradient estimate for $all$ positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise 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 Li-Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on th…

Remarks on curvature behavior at the first singular time of the Ricci flow

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, using Perelman's $\mathcal{W}$-functional, we show that suitable blow-ups of our evolving metrics converge in the pointed Cheeger-Gromov sense to a Gaussian shrinker. If the flow has uniformly bounded scalar curvature and develops Type II singularities at $T$, we show that suitable…

拓扑量子引力中的 Perelman Ricci 流

Perelman's Ricci Flow in Topological Quantum Gravity

该研究在拓扑非相对论量子引力中找到使 Perelman Ricci 流成为路径积分局域化方程的区域，并建立熵泛函、τ 函数与场论结构的对应。

Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which we describe. This analysis applies in particular to the ancient solution constructed by Perelman.

A renormalized Perelman-functional and a lower bound for the ADM-mass

In the first part of this short article, we define a renormalized F-functional for perturbations of non-compact steady Ricci solitons. This functional motivates a stability inequality which plays an important role in questions concerning the regularity of Ricci-flat spaces and the non-uniqueness of the Ricci flow with conical initial data. In the second part, we define a geometric invariant lambda_AF for asymptotically flat manifolds with nonnegative scalar curvature. This invariant gives a quantitative lower bound for the ADM-mass from general relativity, motivates a Ricci flow proof of the r…

Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries

We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.

Ancient Ricci flows with asymptotic solitons

We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman. We prove that the asymptotic solitons must coincide with Bamler's tangent flows at infinity. Furthermore, we show that Perelman's $ν$-functional is uniformly bounded on such ancient solutions; this fact leads to logarithmic Sobolev inequalities and Sobolev inequalities. Lastly, as an important tool for the proofs of the above results, we also show that, for a complete Ricci flow with bounded curvature, the bound of the Nash entropy depends only on the local geometry around an $H_n$-center of its base point.

The Sasaki-Ricci flow and compact Sasakian manifolds of positive transverse holomorphic bisectional curvature

We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler-Ricci flow to the Sasakian setting. We also show that the positivity of transverse of holomorphic bisectional curvature is preserved along the flow, using the methods and the results proved by Bando and Mok in Kahler setting. In particular, we show that the Sasaki-Ricci flow wou…

Some results for the Perelman LYH-type inequality

Let $(M,g(t))$, $0\le t\le T$, $\partial M\neϕ$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without u…

Scalar curvature, entropy, and generalized Ricci flow

We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles.

Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelm…

Ricci flow on open 3-manifolds and positive scalar curvature

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1 or to some mem- ber of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a variant of Perelman's surgery construction for Ricci flow.

A note on blowup limits in 3d Ricci flow

We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.