Some properties of the Schouten tensor and applications to conformal geometry

Left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor

In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure constants of corresponding Lie algebras.

Osserman manifolds and Weyl-Schouten Theorem for rank-one symmetric spaces

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in $\mathbb{R}^16$. As a consequence, we show that a Riemannian manifold having the same Weyl tensor a…

Rigidity of Bach-flat gradient Schouten solitons

In this paper we show that a complete Schouten soliton whose Ricci tensor has at most two eigenvalues at each point is rigid. This allows the classification of both shrinking and expanding Bach-flat Schouten solitons for $n\geq$ 4. When $n=3$ we are able to conclude rigidity under a more general condition, namely when the Bach tensor is divergence free. These results imply rigidity of locally conformally flat Schouten solitons for $n\geq3$.

Weyl-Schouten Theorem for symmetric spaces

Let N be a symmetric space of dimension n > 5 whose de Rham decomposition contains no factors of constant curvature and let W be the Weyl tensor of N at some point. We prove that a Riemannian manifold whose Weyl tensor at every point is a positive multiple of W is conformally equivalent to N (the case N = R^n is the Weyl-Schouten Theorem).

Existence and uniqueness of Green's functions to nonlinear Yamabe problems

For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theore…

An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.

Weighted renormalized volume coefficients

We define weighted renormalized volume coefficients and prove that they are variational. We also prove that they can be written as polynomials of weighted extended obstruction tensors, the weighted Schouten tensor, and the weighted Schouten scalar.

共形几何中的p-Laplace方程

p-Laplace equations in conformal geometry

本文在共形几何中引入中间Schouten曲率的p-Laplace方程，利用p-超调和函数与非线性位势理论得到奇异集Hausdorff维数估计和同伦群消失等几何拓扑结果，推广了Schoen-Yau的工作。

On the partial uniform ellipticity and complete conformal metrics with prescribed curvature functions on manifolds with boundary

We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.

关于完备度量共形类中预定问题的推广

An extension of prescribed problems on the conformal classes of complete metrics

本文在光滑完备度量的共形类中解决了修正 Schouten 张量的预定问题，通过验证算子的均匀椭圆性推广了先前结果，并证明所用假设在拓扑和函数论意义下是精确的。

Extended obstruction tensors and renormalized volume coefficients

The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors". These together with the Schouten tensor constitute building blocks for the coefficients in the ambient metric expansion. The renormalized volume coefficients have recently been considered by Chang-Fang motivated by comparison with the elementary symmetric functions of the eigen…

The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

五维幂零李群上的类Schouten度量

Schouten like metrics on five dimensional nilpotents Lie groups

本文针对预定里奇曲率问题引入类Schouten度量，通过联系代数Schouten孤子，对五维幂零李群上的此类度量及nilsolitons进行分类，深化其几何理解。

Geometric inequalities on locally conformally flat manifolds

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

Variational status of a class of fully nonlinear curvature prescription problems

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

关于带边流形上负曲率共形度量的注记

Notes on conformal metrics of negative curvature on manifolds with boundary

本文通过Morse函数构造共形度量，使修正Schouten张量特征值落入给定锥，从而证明紧致带边3-流形上的任意黎曼度量都共形于负截面曲率度量。

A class of variational functionals in conformal geometry

We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of the metric on the manifold.

A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.

共形Walker度量与线性Fefferman-Graham方程

Conformal Walker metrics and linear Fefferman-Graham equations

针对共形几何中Ricci平坦环境度量例子少的问题，本文基于两步幂零Schouten张量度量，提出方法得到显式环境度量，含左不变Walker度量和pp波，扩展了已知例子。

Conformal deformations of the smallest eigenvalue of the Ricci tensor

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the the…

共形里奇曲率与谱估计

Conformal Ricci Curvature and Spectral Estimates

该研究针对Shen和Ye的共形里奇张量，推导了有界区域第一狄利克雷特征值的优化上界，还得到完备非紧流形底谱等相关精确估计，实现了拉普拉斯算子与共形拉普拉斯算子谱的精确比较。

Classification of singular radial solutions to the $σ_k$ Yamabe equation on annular domains

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE. However, the singular behavior of these solutions, which is important in describing many important questions in conformal geometry, is little u…

闭合负曲率流形上的非线性给定曲率问题

Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature

本文研究了闭合负曲率流形上基于修改Schouten张量的非线性给定曲率问题，证明在满足特定结构条件时，此类问题总能被解决。

Invariant conformal metrics on S^n

In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them \emph{isoparametric …

关于在共形几何中全非线性方程奇异解的粘合构造

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

本文通过研究$σ_2$-Yamabe方程的奇异解构造，展示了经典粘合方法在全非线性设定中的应用，利用共形性质保证线性化算子在加权空间中的良好映射性质。

Conformal Riemannian P-Manifolds with Connections whose Curvature Tensors are Riemannian P-Tensors

The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kähler tensors in Hermitian geometry.

Killing and Conformal Killing tensors

We introduce an appropriate formalism in order to study conformal Killing (symmetric) tensors on Riemannian manifolds. We reprove in a simple way some known results in the field and obtain several new results, like the classification of conformal Killing $2$-tensors on Riemannian products of compact manifolds, Weitzenböck formulas leading to non-existence results, and construct various examples of manifolds with conformal Killing tensors.

A note on the coincidence of the projective and conformal Weyl tensors

This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einstein metric.

The geometry of conformally Einstein metrics with degenerate Weyl tensor

The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations arising on conformally Einstein spaces (with Riemannian signature) where this condition fails, which then allows us to characterize a general class of locally conformally Einstein Riemannian manifolds with degenerate Weyl tensor.

Conformally invariant Cotton and Bach tensor in N-dimensions

This paper presents conformal invariants for Riemannian manifolds of dimension greater than or equal to four whose vanishing is necessary for a Riemannian manifold to be conformally related to an Einstein space. One of the invariants is a modification of the Cotton tensor, the other is a $n$--dimensional version of the Bach tensor. In general both tensors are smooth only on an open and dense subset of $M$, but this subset is invariant under conformal transformations. Moreover, we generalize the main result of "Conformal Einstein Spaces in $N$--Dimensions" published in \emph{Ann. Global Anal. G…

通过双曲几何实现共形平坦结构

Conformally flat structures via hyperbolic geometry

本文研究了双曲空间中浸入结构的共形平坦性，证明了双曲空间的理想边界与浸入结构之间存在对偶关系，并给出了任意维度下的对应方程及唯一解的条件。

The ambient obstruction tensor and conformal holonomy

For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurow…

Prescribed curvature tensor in locally conformally flat manifolds

Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifolds. As applications of these results we exhibit explicit examples of metrics g, conformal to g they are solutions for this problem.

On the global structure of conformal gradient solitons with nonnegative Ricci tensor

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product $\mathbb{R}\times N^{n-1}$, or globally conformally equivalent to the Euclidean space $\mathbb{R}^{n}$ or to the round sphere $\mathbb{S}^{n}$. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. A symmetric tensor field is a trace-free field if the contraction of the field with the metric tensor is identically equal to zero. On a Riemannian manifold of dimension at least three, the space of trace-free conformal Killing symmetric tensor fields of …

M-eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds

We generalized Xiang, Qi and Wei's results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M-eigentriple has been found. We also discussed the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We proved that the M-eigenvalue can determine the Riemann curvature tensor uniquely and generalize the real M-eigenval…

Cotton tensor and conformal deformations of three-dimensional Ricci flow

We study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of Cotton-York tensor and the L1-norm of it under the Ricci flow. In particular, we investigate the behavior of the L1-norm of the Cotton-York tensor under the Ricci flow on three-dimensional simply-connected homogeneous manifolds which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the L1-norm of the Cotton-York tensor for the product metric of the Rosenau solution for the Ricci flow on a two-sphere and the standard metric of a cir…

共形Killing张量及其Killing尺度

Conformal Killing tensors and their Killing scales

本文通过共形不变的tractor演算和微分延拓，刻画了共形Killing张量成为Killing张量无迹部分的条件，给出了Einstein Killing尺度的代数刻画，并研究了共形Killing向量的类似问题。

Integrability Conditions for Killing-Yano Tensors and Conformal Killing-Yano Tensors

The integrability conditions for the existence of a conformal Killing-Yano tensor of arbitrary order are worked out in all dimensions and expressed in terms of the Weyl tensor. As a consequence, the integrability conditions for the existence of a Killing-Yano tensor are also obtained. By means of such conditions, it is shown that in certain Einstein spaces one can use a conformal Killing-Yano tensor of order p to generate a Killing-Yano tensor of order (p-1). Finally, it is proved that in maximally symmetric spaces the covariant derivative of a Killing-Yano tensor is a closed conformal Killing…

共形紧度量与 Lovelock 张量

Conformally compact metrics and the Lovelock tensors

本文研究满足推广Einstein方程的Lovelock方程的共形紧度量，证明其容许多重齐次展开以实现Fefferman-Graham展开，并在偶数维中识别出光滑性边界障碍，同时构造了奇异Yamabe-(2q)问题的形式解及自旋流形填充的指标障碍。

The conformal deformation detour complex for the obstruction tensor

On pseudo-Riemannian manifolds of even dimension $n\geq 4$, with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.

Conformal Killing-Yano Tensors on Manifolds with Mixed 3-Structures

We show the existence of conformal Killing-Yano tensors on a manifold endowed with a mixed 3-Sasakian structure.

The spectral geometry of the Weyl conformal tensor

We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.

李群上的共形Killing对称张量

Conformal Killing symmetric tensors on Lie groups

该研究提出Killing型度量李代数的概念，证明满足两步幂零、低维等多种条件的李代数关于任意正定度量均为Killing型，丰富了李群上共形Killing对称张量的相关理论。

Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime

The higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic equations and Klein-Gordon equations. We present a classification of higher-dimensional spacetimes admitting a rank-2 closed CKY tensor. This provides a generalization of the Kerr-NUT-de Sitter spacetime. In particular, we show that the Kerr-NUT-de Sitter spacetime is the only s…

齐性平面波上的不可约 Killing 张量与共形 Killing 张量

Irreducible Killing and conformal Killing tensors on homogeneous plane waves

该研究借助 BGG 算子分类齐性平面波上的不可约 Killing 与共形 Killing 2-张量，并给出显式公式及存在条件。

On the characterization of non-degenerate foliations of pseudo-Riemannian manifolds with conformally flat leaves

A necessary and sufficient condition for the leaves of a {\em non-degenerate} foliation of a pseudo-Riemannian manifold to be conformally flat is developed. The condition mimics the classical condition of the vanishing of the Weyl or Cotton tensor establishing the conformal flatness of a pseudo-Riemannian manifold in the sense that it is also formulated in terms of the vanishing of certain tensors. These tensors play the role of the Weyl or the Cotton tensors and they are defined in terms of the the curvature of a linear torsion-free connection (the {\em bi-conformal connection}).

The ambient obstruction tensor and the conformal deformation complex

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor lea…