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.     

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.     

On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature, i.e., \(\delta W=0\). Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product \( \mathbb {R}^2 \times N_{\lambda }\) of the Euclidean metric and a 2-d Riemannian manifold of constant curvature \({\lambda } \ne 0\), a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao–Chen’s works (in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013) and Derdziński’s study on Codazzi tensors (in Math Z 172:273–280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with \(\delta W=0\). For the shrinking case, it re-proves the rigidity result (Fernández-López and García-Río in Math Z 269:461–466, 2011; Munteanu and Sesum in J. Geom Anal 23:539–561, 2013) in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally flat ones with \(\delta W=0\). We also characterize locally 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.     

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.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

Steady Ricci flows

Steady solutions for Ricci flows are given. A class of Riemannian 3-manifolds related to the geometry of a surface is considered. The components of the metric tensor, which reproduce the Riemannian space and a triorthogonal coordinate system, are determined by a system of partial differential equations. In the stationary case, the curvature tensor of the space satisfies six equations determining the metric of the space. The exact analytic solutions corresponding to surfaces of constant Gaussian and mean curvature (n = 3) are written. Arbitrary curvilinear coordinate systems are constructed, on which the construction of structured grids is based.     

Three-Dimensional Homogeneous Generalized Ricci Solitons

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.     

Rigidity and Infinitesimal Deformability of Ricci Solitons

In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.     

Certain Contact Metrics as Ricci Almost Solitons

We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S ²ⁿ⁺¹. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E ⁿ⁺¹ ×  S ⁿ (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.     

在$L^p$条件下完备Ricci孤立子流形的刚性定理

我们证明了在一定曲率和$L^p$条件下完备Ricci孤立子流形的一些刚性结果.     

Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M ⁿ and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.     

Conformally Flat Algebraic Ricci Solitons on Lie Groups

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segrè type {(1... 1 2)} with a unique eigenvalue (equal to 0).     

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.     

Geometric Properties of Self-Shrinkers in Cylinder Shrinking Ricci Solitons

In this paper, we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.     

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny–Kowalski’s types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons.     

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and KobayashiNomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.     

Gradient estimates for the heat equation under the Ricci flow

The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M.     

Gradient estimates of porous medium equations under the Ricci flow

In this paper we consider an n-dimensional manifold M ⁿ evolving under the Ricci flow and establish gradient estimates for positive solutions of porous medium equations on M ⁿ . As applications, we derive Harnack type inequalities. In particular, our results generalize gradient estimates for positive solutions of the heat equations in Liu (Pacific J Math 243:165–180 [18]).     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.