Stuart James Hall
This is a short note in which we show how to calculate the value of Perelman's nu-functional for a variety of metrics. In particular we complete the calculation of values for the known 4-dimensional Einstein and shrinking Ricci soliton metrics.
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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.
In this paper, we develop the theory of Perelman's $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvature satisfies certain condition near the singularities. We also obtain an asymptotic order for the minimizer near the singularities.
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 paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schrödinger operator $-4Δ+R$ consists of discrete eigenvalues with finite multiplicities, if the scalar curvature $R$ satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectral properties, we extend the theory of the Perelman's $λ$-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.
A Spinorial Perelman's Functional: Critical Points and Gradient Flow
本文提出一种统一Perelman泛函、Baldauf-Ouzch泛函和Dirichlet能量的旋量能量泛函，研究其临界点与梯度流的性质。
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。
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.
We show a quite simple second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow over Fano manifolds.
Perelman's functionals on manifolds with non-isolated conical singularities
本文从谱角度在具有非孤立锥形奇点的流形上定义 Perelman 泛函，证明其谱离散性及 W-泛函下确界有限，并研究特征函数渐近行为及更一般奇点情形。
We show a very simple and general total second variation formula for Perelman's $\mathcal{W}$-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for Perelman's $\mathcal{W}$-functional with respect to such variations. In this case the main therm in the formula depends strongly on the variation of the complex structure. We discover also convexity of Perelman's $\mathcal{W}$-functional along particular variations over points wit…
In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's $λ$ and $ν$ functionals of cones and characterize their finiteness in terms of the $λ$-functional of the link. As an application, we characterize manifolds with conical singularities on which a $λ$-functional can be defined and get upper bounds on the $ν$-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theo…
In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized $λ$ constant. The other results are about the existence of ground states of generalized $F$-functional and W-functional both on a complete non-compact Riemannian manifold $(M,g)$ with positive injectivity radius and with Ricci curvature bounded from below. Our main results are Theorems 2,3 and 7. For the existence of the ground states we use Lions' concentration-com…
Perelman's entropy and heat kernel bounds on RCD spaces
本文研究了有限维RCD空间上的Perelman W-熵泛函，严格论证了其时间导数公式并推导单调性与刚性性质，同时建立了具有独立意义的热方程解的界。
In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichnerowicz Laplacian is nonpositive, (B) lambda satisfies a Lojasiewicz-Simon gradient inequality, (C) the Ricci flow does not move excessively in gauge directions. As consequences, we obtain a rigidity result, a new proof of Sesum's dynamical stability theorem, and a dynamical insta…
We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.
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…
Using Perelman's results on Kahler Ricci flow, we prove that the K energy is bounded from below if and only if the F functional is bounded from below in the canonical Kahler class.
This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy $\check{W}^λ$. We regard Perelman's W-entropy as a functional on the tangent bundle $T \mathcal{H} (X, L)$ of the space $\mathcal{H} (X, L)$ of K"ahler metrics in a given K"ahler class $L$. The critical points of $\check{W}^λ$ turn out to be $μ^λ$-cscK metrics. When $λ\le 0$, the supremum along the fibres gives a smooth…
Let $(\mathcal{M},g)$ be a closed Riemannian manifold. The $\textit{ second order approximation}$ to the perturbative renormalization group flow for the nonlinear sigma model (RG-2 flow) is given by : \[ \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm{Ric}(t) \, -\, \fracα{2} \mathrm{Rm}^2(t), \] where $ g = \mathrm{Riemannian \ metric}, \mathrm{Ric} = \mathrm{Ricci \ curvature, } \ \mathrm{Rm}^2_{ij}:=\mathrm{R}_{irmk}\mathrm{R}_j^{rmk},$ and $α\ge 0$ is a parameter. The flow is invariant under diffeomorphisms, but not under scaling of the metric. We first develop a $\textit{geometrica…
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…
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…
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。
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…
In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.
We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $λ$-functional adapted to the ALE situation and denoted $λ_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.
A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification …
We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler-Ricci flow provides weak convergence in…
This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on each minimal projective manifold of general type.
We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant mu-scalar curvature K"ahler metric by investigating a volume functional as a generalization of Tian-Zhu's work, which is closely related to Perelman's W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant mu-scalar curvature and as a prelud…
Ricci Flow on ALF manifolds
本文证明Ricci流保持ALF结构，定义重整化泛函$\lambda_{\mathrm{ALF}}$，建立变分与线性稳定性概念，并证明共形Kähler非超Kähler例子沿Ricci流动态不稳定，且$\lambda_{\mathrm{ALF}}$符号与正相对质量相关。
The Harmonic Variational Principle for the Einstein-Hilbert Functional
本文针对紧致黎曼流形的爱因斯坦-希尔伯特泛函引入受限变分原理，推导其欧拉-拉格朗日方程，证明调和临界度量等价于紧致梯度里奇孤立子的度量，并给出其规范1-形式的性质及非爱因斯坦型调和临界度量的收缩性结论。
We introduce a new functional inspired by Perelman's $λ$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $λ_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $λ_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a…
Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series f…
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