The classification of locally conformally flat Yamabe solitons

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on Rⁿ${\mathbb{R}}^n$ is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabe soliton.     

On complete Finslerian Yamabe solitons

Here it is shown that any Finslerian compact Yamabe soliton with bounded above scalar curvature is of constant scalar curvature. Furthermore, this extension of Yamabe solitons is developed for inequalities and among the others, it is proved that a forward complete non-compact shrinking Yamabe soliton has finite fundamental group and its first cohomology group vanishes, providing the scalar curvature is strictly bounded above.     

On noncompact quasi Yamabe gradient solitons

We study τ-quasi Yamabe gradient solitons on complete noncompact Riemannian manifolds. We prove several scalar curvature estimates under some conditions and get a non-local collapsing result based on the gradient estimate of the potential function. We also derive a decay theorem and a finite topological type result.     

On conformal solutions of the Yamabe flow

The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.     

耗散双曲Yamabe流的精确解

本文介绍了一种新的几何流, 得到了这种流的一些精确解. 首先得到了初始度量为Einstein的解. 其次得到了具有轴对称的解. 最后, 作为这种流的特殊解, 定义了稳定耗散双曲Yamabe孤子, 而且给出了这种孤子解所满足的方程.     

Properties of complete non-compact Yamabe solitons

In this article, we first study the local volume estimate of the complete non-compact Yamabe soliton. Then we study the behavior of the potential function of the steady Yamabe soliton with positive Ricci curvature. We also study the scalar curvature decay of steady and expanding Yamabe solitons with Ricci pinching condition.     

Yamabe and Quasi-Yamabe Solitons on Euclidean Submanifolds

In this paper, we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.     

Remarks on scalar curvature of Yamabe solitons

In this article, we consider the scalar curvature of Yamabe solitons. In particular, we show that, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that a complete non-compact Yamabe soliton with the quadratic decay at infinity of its Ricci curvature has non-negative scalar curvature. A new proof of Kazdan?CWarner condition is also presented.     

Characterization of Almost η-Ricci–Yamabe Soliton and Gradient Almost η-Ricci–Yamabe Soliton on Almost Kenmotsu Manifolds

Acta Mathematica Sinica, English Series - The prime object in this article is to study an almost η-Ricci–Yamabe soliton and gradient almost η-Ricci–Yamabe soliton within the...     

Ricci and Yamabe solitons on second-order symmetric,and plane wave 4-dimensional Lorentzian manifolds

We show that a proper second-order symmetric spacetime, and four-dimensional Lorentzian plane wave manifolds admit different vector fields resulting in expanding, steady and shrinking Ricci and Yamabe solitons. Moreover, it is proved that those Ricci and Yamabe solitons are gradient only in the steady case.     

Yamabe流下黎曼流形上热方程的梯度估计

研究了在Yamabe流下演化的一个完备非紧黎曼流形,对流形上热方程的正解给出了两种局部的梯度估计.作为应用,可以得到这个热方程的Harnack不等式.     

Three-dimensional homogeneous Lorentzian Yamabe solitons

A geometric characterization of Yamabe solitons on homogeneous Lorentzian manifolds of dimension three is given. As a consequence, Lorentzian Yamabe solitons and left-invariant Lorentzian Yamabe solitons are classified in this setting, showing the existence of Yamabe solitons which are not left-invariant.     

On the CR Poincaré–Lelong Equation,Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds

In this paper, we solve the so-called CR Poincaré–Lelong equation by solving the CR Poisson equation on a complete noncompact CR(2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of K?hler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.     

The contact Yamabe flow on K-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.     

On Three-Dimensional CR Yamabe Solitons

In this paper, we investigate the geometry and classification of three-dimensional CR Yamabe solitons and pseudo-gradient CR Yamabe solitons. In the compact case, we obtain a classification result of three-dimensional CR Yamabe solitons under the assumption that their potential functions are in the kernel of the CR Paneitz operator. In addition, we obtain a structure theorem on the diffeomorphism types of complete three-dimensional pseudo-gradient CR Yamabe solitons (shrinking, steady, or expanding) of vanishing torsion.     

Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension

Motivated by the definition of combinatorial scalar curvature given by Cooper and Rivin, we introduce a new combinatorial scalar curvature. Then we define the discrete quasi-Einstein metric, which is a combinatorial analogue of the constant scalar curvature metric in smooth case. We find that discrete quasi-Einstein metric is critical point of both the combinatorial Yamabe functional and the quadratic energy functional we defined on triangulated 3-manifolds. We introduce combinatorial curvature flows, including a new type of combinatorial Yamabe flow, to study the discrete quasi-Einstein metrics and prove that the flows produce solutions converging to discrete quasi-Einstein metrics if the initial normalized quadratic energy is small enough. As a corollary, we prove that nonsingular solution of the combinatorial Yamabe flow with nonpositive initial curvatures converges to discrete quasi-Einstein metric. The proof relies on a careful analysis of the discrete dual-Laplacian, which we interpret as the Jacobian matrix of curvature map.     

Compactness of solutions to the Yamabe problem. III

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.     

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n ? 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.     

Weak and smooth solutions for a fractional Yamabe flow: The case of general compact and locally conformally flat manifolds

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant fractional curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators.