Sigurd Angenent, Dan Knopf
The best known finite-time local Ricci flow singularity is the neckpinch, in which a proper subset of the manifold becomes geometrically close to a portion of a shrinking cylinder. In this paper, we prove precise asymptotics for rotationally symmetric Ricci flow neckpinches. We then compare these rigorous results with formal matched asymptotics for fully general neckpinch singularities.
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In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. This is the first example of Ricci flow starting from a type-II singularity.
Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.
Exponential Asymptotics of Ricci Flow Neckpinch
本文研究紧致流形上 Ricci 流的柱形颈部收缩奇点，在 Type-I 假设下证明先前构造的渐近轮廓是唯一的可能。
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion components. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forwa…
In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer $k\geq3$, there exist compact solutions in all dimensions $m\geq3$ that become singular at the rate (T-t)^{-2+2/k}$.
In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $\mathbb{S}^{n+1}$. In particular, we recover their proof of 'single-point pinching' along the flow. Using the methods of optimal transportation, we are able to remove the assumption of reflection symmetry for the metric. Our argument relies on the heuristic for weak Ricci flow proposed by McCann-Topping which characterizes super solutions of the Ricci flow by the contractivity of diffusions.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that t…
We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber $\{s\}\times\mathrm{SU}(2)$ a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch singularities, and that they asymptotically approach round product metrics in space-time neighborhoods of their singular sets, in precise senses. These are the first examples of Ricci flow solutions without rotational symmetry that become asymptotically rotationally symmetric loc…
In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Flat (SWIF) distance.
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".
We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singularity formation. Our results show in a precise way that MCF solutions become asymptotically rotationally symmetric near a neckpinch singularity.
Formation and structural stability of nondegenerate neckpinches
本文证明平均曲率流中非退化颈缩奇点在有限时间内形成，给出其精确渐近轮廓，并证明其在C^2扰动下结构稳定，结合密度定理表明该现象在三维中普遍且稳定。
We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension $3$ a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of sing…
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological…
It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity deve…
Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these eq…
An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this short note, we show that on a compact manifold, the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, without any hypotheses on the initial data.
Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle
本文证明当锥角趋于零时，锥形凯勒-里奇流收敛到唯一的凯勒-里奇流，该流在除子外光滑并沿除子具有尖点奇异性。
Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
该研究证明在闭合流形上存在一个里齐流解，其在有限时间形成基于渐近圆锥收缩子流形的奇异性和模型。
Static Vacuum Spacetimes with $Λ<0$ as Attractors of the Ricci-Harmonic Flow
研究了Λ<0的静态渐近超几何解的动态稳定性，证明了正质量定理与Ricci-谐波流的稳定性之间的关系。
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…
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ri…
Along the Ricci flow, we study the polyhomogeneity of complete Riemannian metrics endowed with "a Lie structure fibred at infinity", that is, a class of Lie structures at infinity that induce in a precise way a fibre bundle structure on a certain compactification by a manifold with corners. When the compactification is a manifold with boundary, this class of metrics contains, in particular, the b-metrics of Melrose, the fibred boundary metrics of Melrose and Mazzeo and the edge metrics of Mazzeo. Our main result consists in showing that the polyhomogeneity of the metrics compatible with a Lie …
We consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^2} u_0 $. Assuming that $u_0$ is compactly supported we describe precisely the Type II vanishing behavior of $u$ at time $T$: we show the existence of an inner region with exponentially fast vanishing profile, which is, up to proper scaling, a {\em soliton cigar solution}, and th…
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.
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。
In this paper we analyze the long-time behaviour of 3 dimensional Ricci flow with surgery. We prove that under the topological condition that the initial manifold only has non-aspherical or hyperbolic components in its geometric decomposition, there are only finitely many surgeries and the curvature is bounded by $C t^{-1}$ for large $t$. This answers an open question in Perelman's work, which was made more precise by Lott and Tian, for this class of initial topologies. More general classes will be discussed in subsequent papers using similar methods.
We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies in particular to the solution constructed by G.Perelman
We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.
On the Ricci flow on Trees
本文研究了有限树上基于Lin-Lu-Yau版本的Ollivier里齐曲率的里齐流，证明只有caterpillar树才能收敛到边曲率为零的度量。
We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its omega-limit set consists of a sing…
Unique Asymptotics of Steady Ricci Solitons with Symmetry
本文研究具有 O(3)-对称性的四维梯度稳态 Ricci 孤立子，在弱曲率衰减条件下得到其精确几何渐近性，为分类问题奠定基础。
We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. And we prove that a related characterization is equivalent to an asymptotic lower bound on the growth of the Wasseretein distance between heat flows. For weighted Riemannian manifolds, the crucial result will be a precise uniform two-sided bound for \begin{eqnarray*}\frac{d}{dt}\B…
Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows
本文证明了在数值有效流形上，Kähler-Ricci流解的奇点类型不依赖于初始度量的选择，推广了Y. Zhang在半丰富情形下的结果。
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 …
Finite Time Type I Singularities of the Kähler Ricci Flow
该研究证明了Kähler-Ricci流有限时间I型奇点的Feldman--Ilmanen--Knopf猜想，确定了对应抛物爆破极限的具体形式。
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$ Δ- R(x,t)$", where $R$ is the scalar curvature of the Ricci flow, which is well generalised to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by product, we obtain some Li-Yau type…
In the vector space of algebraic curvature operators we study the reaction ODE $$\frac{dR}{dt} = R^2+R^{#}= Q(R)$$ which is associated to the evolution equation of the Riemann curvature oper- ator along the Ricci flow. More precisely, we analyze the stability of a special class of zeros of this ODE up to suitable normalization. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces $M \times \mathbb{R}^k, \ k \geq 0$ where $M$ is an Einstein (locally) symmetric space of compact type and not a spherical space form when $k = 0.$
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immerse…
This is the second part of a series of papers analyzing the long-time behaviour of 3 dimensional Ricci flows with surgery. We generalize the methods developed in the first part and use them to treat cases in which the initial manifold satisfies a certain purely topological condition which is far more general than the one that we previously had to impose. Amongst others, we are able to treat initial topologies such as the 3-torus or $Σ\times S^1$ where $Σ$ is any surface of genus $\geq 1$. We prove that under this condition, only finitely many surgeries occur and that after some time the curvat…
We study the Ricci flow on $\mathbb{R}^{n+1}$, with $n\geq 2$, starting at some complete bounded curvature rotationally symmetric metric $g_{0}$. We first focus on the case where $(\mathbb{R}^{n+1},g_{0})$ does not contain minimal hyperspheres; we prove that if $g_{0}$ is asymptotic to a cylinder then the solution develops a Type-II singularity and converges to the Bryant soliton, while if the curvature of $g_{0}$ decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotica…
We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension n with a q-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solu…
On the space of harmonic forms and Schrödinger heat equations on noncompact manifolds
本文在允许里奇曲率为负的流形上，在尺度不变Sobolev不等式、弱Hardy不等式及小系数二次曲率衰减条件下，证明多项式增长调和1-形式空间有限维，并推广至薛定谔热方程解。
In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the mean curvature flow on $M^n \times [0,T]$ with bounded second fundamental forms. Suppose $F_T=\widetilde{F}_T$, then $F_t=\widetilde{F}_t$ on $M^n \times [0,T]$. This is an analog of a result of Kotschwar on the Ricci flow.
Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow
本文在广义圆柱体假设下扩展了Ricci流中指态W熵的Lojasiewicz不等式，证明了广义圆柱切流的强唯一性，并展示了由某些切流给出R^k×N^{n−k}或其商的子集S^k_qc(N)是水平抛物线k-可测的。
Curvature estimates for four-dimensional complete gradient expanding Ricci solitons
本文研究四维完备非负Ricci曲率梯度扩张Ricci孤立子，得到曲率张量及其协变导数由标量曲率界定的估计式，并应用于证明有限渐近标量曲率比蕴含有限渐近曲率比与$C^{1,α}$渐近锥。
We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a compactness theorem in space of the kahler metrics given unifrom Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-sigularity result about a weak constant scalar curveture metric in a punctured ball. One of the main difficulties is that the metr…
Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three
研究从闭可定向三维黎曼流形出发的奇异 Ricci 流在自然时空距离下的完备化，证明其时间片具有由共轭热核测度定义的规范内蕴度量，且这些时间片作为流正则部分的度量极限出现，并建立了完备化奇异集的结构理论。
Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits
本文研究紧凯勒流形上有限时间非坍缩凯勒-里奇流的热核估计，构造其格罗莫夫-豪斯多夫极限，证明固定基点I型爆破的子序列度量流极限为奇异收缩凯勒-里奇孤子空间，移动基点的极限度量流在熵挤压条件下为度量孤子。
Monotonicity of the Liouville entropy along the Ricci flow on surfaces
研究显示，在非常数负曲率闭合曲面上，规范里齐流下里奥维勒熵最终严格递增，并推导出二维任一共形变形下里奥维勒熵导数的新表达式，证明其在1/6弯曲度指标方向上为正，部分回答了曼宁2004年的问题。
Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
本文研究了凯勒-里茨流中Fano纤维的几何结构，推导了直径界和标量曲率估计，并给出了复蒙日-安培流势的精确估计。
Mean Curvature Flow and Heegaard Surfaces in Lens Spaces
研究了完备非负各向同性曲率 3 维流形中 mean convex 两球的模空间连通性，证明其路径连通，并探讨了 Heegaard 扭曲面的路径连通分支数与其 ambient 曼尼福德同调类型的关系。
Unique asymptotics of $SO(k)\times SO(n-k+1)$ symmetric ancient ovals of Ricci flow
研究\(n\geq4\)且\(2\leq k\leq n - 2\)时Ricci流的\(SO(k)\times SO(n - k + 1)\)不变、紧致、非自相似\(\kappa\)-解，获其唯一渐近性，给出相关结果并证明唯一性。
Forman--Ricci Curvature on Contact-Sequence Temporal Networks via Spatiotemporal Prism Complexes
本文提出一种几何框架，通过棱柱构造将接触序列时间网络提升为真正的单纯复形，并开发了Forman-Ricci曲率，对比了两种变体，证明其在均匀权重下一致，并通过数值实验验证了理论预测。
Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to $M^\times$ (more precisely, it carries a smooth structure à la Gompf) then $X^\times$ can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently $X^\times$ with this metric is in fact a hyper-K…
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。
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