The contact Yamabe flow on <Emphasis Type="Italic">K</Emphasis>-contact manifolds

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

On the $h$-Almost Yamabe Soliton

We introduce the concept $h$-almost Yamabe soliton which extends naturally the almost Yamabe soliton by Barbosa-Ribeiro and obtain some rigidity results concerning $h$-almost Yamabe solitons. Some condition for a compact $h$-almost Yamabe soliton to be a gradient soliton is also obtained. Finally, we give some characterizations for a special class of gradient $h$-almost Yamabe solitons.     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

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.     

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.     

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.     

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.     

The second Yamabe invariant

Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

Equivariant Yamabe problem and Hebey-Vaugon conjecture

In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey-Vaugon conjecture in dimensions less or equal to 37.     

Discrete Morse Flow for Yamabe Type Heat Flows

In this paper, we study the discrete Morse flow for either Yamabe type heat flow or nonlinear heat flow on a bounded regular domain in the whole space. We show that under suitable assumptions on the initial data $g$ one has a weak approximate discrete Morse flow for the Yamabe type heat flow on any time interval. This phenomenon is very different from the smooth Yamabe flow, where the finite time blow up may exist.     

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.     

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

Perturbation of the CR fractional Yamabe problem

In this note, we first prove the non‐degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni 30 .     

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.     

Sur quelques problèmes de courbure scalaire

In this article we prove, among other things, some results about two problems which are the subject of announces these last decades: (1) the compactness of the set of the solutions of the Yamabe equation on a compact Riemannian manifold, (2) a generalization of a result of the author which is necessary to solve the Yamabe problem, when 2ω?n−6.     

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption, we derive a formula for the derivative of Yamabe constant , where g(t) is a solution of Ricci flow on closed manifold. We also give a simple application. Mathematics Subject Classifications (2000): 53C21 and 53C44     

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.     

Yamabe不变量与在非紧流形上预定纯量曲率

我们得到关于Ｙａｍａｂｅ不变量的某些条件,使得非紧流形通过共形形变后,纯量曲率为常数．