arXiv math/05093412005-09-15math.DG

On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

Neil S. Trudinger, Xu-Jia Wang

备注：: 22 pages

英文摘要：

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 <k<n$, we prove a sharp Harnack inequality for admissible metrics when $(M,g_0)$ is not conformally equivalent to the unit sphere $S^n$ and that the set of all such metrics is compact. When $(M,g_0)$ is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering a recent result of Gursky and Viaclovski on the solvability of the $k$-Yamabe problem for $k>n/2$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A fully nonlinear version of the Yamabe problem and a Harnack type inequality

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

2002-12-02math.APmath.DG

On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe

The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.

2004-03-25math.APmath.DG

The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3

We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

2000-05-24math.CVmath.DG

Harnack Inequalities for Yamabe Type Equations

We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.

2006-04-25math.APmath.DG

关于$k$-Yamabe梯度孤立子的存在性与分类

On the existence and classification of $k$-Yamabe gradient solitons

本文研究$k$-Yamabe流的旋转对称共形平坦容许孤立子，通过分析相关动力系统，分维度情况证明不同类型孤立子的存在性与否并刻画渐近行为，完成分类。

2024-10-10math.DG

The Sasaki Join and Admissible Kähler Constructions

We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible Kähler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit solutions to the CR Yamabe problem. In this regard we give examples of the lack of uniqueness when the Yamabe invariant is positive.

2015-06-03math.DG

更多相关论文收起更多论文

Asymptotic behavior of solutions to the $σ_k$-Yamabe equation near isolated singularities

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equa…

2009-11-02math.APmath.DG

The Yamabe problem for higher order curvatures

Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k=1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k>n/2. In this paper we prove the existence of solutions for the remaining cases k <n/2, k=n/2, assuming t…

2005-05-23math.APmath.DG

Volume comparison and the sigma_k-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the k-th elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.

2003-10-19math.APmath.DG

关于$\sigma_2$-Yamabe方程的爆破现象

Blow-up phenomena for the $σ_2$-Yamabe equation

Wuhan University（武汉大学）；Henan University（河南大学）；Chinese University of Hong Kong（香港中文大学）

研究针对$n\geq27$时$\mathbb{S}^n$上特定光滑度量，解决归一化$\sigma_2$-Yamabe方程相关问题。通过构造特定度量，克服线性化算子椭圆性可能丧失的困难，给出有限维约化中的四次轮廓确定端点维数$n = 27$。

2026-07-14math.APmath.DG

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global …

2019-01-14math.APmath.DG

紧埃尔米特流形上对数逆迹流的比较原理与动力学稳定性

Comparison principles and dynamical stability for a logarithmic inverse-trace flow on compact Hermitian manifolds

Chongqing University of Technology（重庆理工大学）

该研究针对紧埃尔米特流形上的对数逆迹流，建立比较原理与相关估计，证明其长时间存在性、指数收敛性，并推导收敛准则以推广动力学稳定性。

2026-08-11math.APmath.DG

完全非线性方程的结构及其在完备共形度量预设问题中的应用

The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

该研究探讨完全非线性方程的结构，引入Morse理论在弱条件下构造容许度量，求解两类几何问题的完全非线性版本，并通过拓扑障碍验证结构条件的最优性。

2025-03-24math.APmath.DG