Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

We give a simple proof of an extension of the existence results of Ricci flow of Giesen and Topping (2010, 2011) [15], [20], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi’s existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of Perelman [7] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of Topping (2010) [16] without using the existence theorem of Shi (1989) [9].     

Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.     

关于二维球面上的Ricci流的一个注记

给出Hamilton和Chow关于二维球面上的Ricci流的一个定理的新证明．这个证明结合了Hamilton关于Ricci流的紧性定理和二维ancientp解的分类和Perelman的非塌缩定理．     

A note on uniformization of Riemann surfaces by Ricci flow

We clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

     

Ricci Flow on Surfaces with Conical Singularities

This paper studies the normalized Ricci flow on surfaces with conical singularities. It’s proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear drop and the football, it’s shown that they admit a Ricci soliton metric.     

Heat kernel bounds, ancient κ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.     

Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds*

In the paper, the authors provide a new proof and derive some new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier results in the literature. And some parabolic type Liouville theorems for ancient solutions are obtained.     

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

A heat semigroup approach to concentration on the sphere and on a compact Riemannian manifold

We give a simple proof of the Lévy concentration of measure phenomenon on the sphere and on a compact Riemannian manifold with strictly positive Ricci curvature using the heat semigroup and Bochner's formula.     

Ricci flow on homogeneous spaces with two isotropy summands

We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of spaces. Moreover, we investigate the existence of ancient solutions and relate this to the existence and non-existence of invariant Einstein metrics.     

Remarks on Kähler Ricci Flow

We show the convergence of Kähler Ricci flow directly if the α-invariant of the canonical class is greater than \(\frac{n}{n+1}\). Applying these convergence theorems, we can give a Kähler Ricci flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of KE metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by Tian (Invent. Math. 89(2):225–246, 1987; Invent. Math. 101(1):101–172, 1990; Invent. Math. 130:1–39, 1997). However, a new proof based on Kähler Ricci flow should be still interesting in its own right.     

Ricci Yang-Mills flow on surfaces

We study the behavior of the Ricci Yang-Mills flow for U(1) bundles on surfaces. By exploiting a coupling of the Liouville and Yang-Mills energies we show that existence for the flow reduces to a bound on the isoperimetric constant or the L⁴ norm of the bundle curvature. We furthermore completely describe the behavior of long time solutions of this flow on surfaces. Finally, in Appendix A we classify all gradient solitons of this flow on surfaces.     

Line congruences as surfaces in the space of lines

The space of lines in R³ can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.     

On the long-time behavior of type-III Ricci flow solutions

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton’s compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of an -action on a compact space. This work was supported by NSF grant DMS-0306242     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.     

Ricci solitons: the equation point of view

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.     

Geometry of Ricci Solitons

Ricci solitons are natural generalizations of Einstein metrics on one hand, and are special solutions of the Ricci flow of Hamilton on the other hand. In this paper we survey some of the recent developments on Ricci solitons and the role they play in the singularity study of the Ricci flow.