On Gradient Ricci Solitons

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons in Math. Z. (2011) classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.     

Maximum principles and gradient Ricci solitons

It is shown that the Omori-Yau maximum principle holds true on complete gradient shrinking Ricci solitons both for the Laplacian and the f-Laplacian. As an application, curvature estimates and rigidity results for shrinking Ricci solitons are obtained. Furthermore, applications of maximum principles are also given in the steady and expanding situations.     

On the essential spectrum of complete non-compact manifolds

In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0,+∞) on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative Ricci curvature in many ways.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

关于完备收缩的Ricci-harmonic孤子的研究

本文研究了收缩的Ricci-harmonic孤子的几何性质的问题.利用文献[4]在Ricci孤子下的方法,获得了每个紧致Ricci-harmonic孤子是一个梯度孤子的结论,推广了Perelman等人在Ricci孤子下的结果.此外,利用文献[14]在Ricci孤子下的方法,获得了完备非紧梯度收缩的Ricci-harmonic孤子具有比至多欧氏增长更加精确的体积增长估计的结果,推广了文献[14]在Ricci孤子下的结果.     

A Myers-type theorem and compact Ricci solitons

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons.

     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

Parallel tensors and Ricci solitons in <Emphasis Type="Italic">N</Emphasis> (<Emphasis Type="Italic">k</Emphasis>)-quasi Einstein manifolds

The Eisenhart problem of finding parallel and symmetric tensors is considered in the framework ofN (k)-quasi Einstein manifolds and the result is connected with Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is: i) shrinking on a class of conformally flat perfect fluid space-times and on quasi-umbilical hypersurfaces, in particular unit spheres; ii) expanding if the generator is of torse-forming type.     

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153?C201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional ${\mathcal{J}}$ associated to a smooth function ${\phi}$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.     

Rigidity theorem for integral pinched shrinking Ricci solitons

We prove that an n-dimensional, \(n\ge 4\), compact gradient shrinking Ricci soliton satisfying a \(L^{\frac{n}{2}}\)-pinching condition is isometric to a quotient of the round \(\mathbb {S}^n\), which improves the rigidity theorem given by Catino (Integral pinched shrinking Ricci solitons, 2016), in dimension \(4\le n\le 6\).     

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

Three-dimensional Lorentzian homogeneous Ricci solitons

We study three-dimensional Lorentzian homogeneous Ricci solitons, proving the existence of shrinking, expanding and steady Ricci solitons. For all the non-trivial examples, the Ricci operator is not diagonalizable and has three equal eigenvalues.     

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.     

Computing Perelman?s ν-functional

This is a short note in which we show how to calculate the value of Perelman?s ν-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.     

Complete shrinking Ricci solitons have finite fundamental group

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of García-Río and Fernández-López in the compact case.

     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

Lorentz Ricci Solitons on 3-dimensional Lie groups

The three-dimensional Heisenberg group H ₃ has three left-invariant Lorentzian metrics g ₁, g ₂, and g ₃ as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g ₁ as a Lorentz Ricci Soliton. This Ricci Soliton g ₁ is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.