Jie Qing
In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions.
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In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{é}-Einstein manifold with the round sphere as its conformal infinity.
Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.
In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.
A Riemannian or pseudo-Riemannian (or conformal) structure is conformally Einstein if and only if there is a suitably generic parallel section of a certain vector bundle -- the so-called standard conformal tractor bundle. We show that this characterisation leads to a systematic approach to constructing obstructions to conformally Einstein metrics. Relaxing the requirement that the parallel tractor field be generic gives a natural generalisation of the Einstein equations.
A Rigidity theorem on conformally compact Einstein manifolds in high dimensions
本文在高维情况下建立了 conformally compact Einstein 流形的刚性定理，扩展了四维结果。
In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group of $\mathbb{S}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant Möbius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $$\mathbb{ S}^1(\sqrt{1-r^2})…
We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the $Q$-curvature. We show how all the global information come from the Scattering.
In the article we introduce new conformal and smooth invariants on compact, oriented four-manifolds with boundary. In the first part, we show that "positivity" conditions on these invariants will impose topological restrictions on underlying manifolds with boundary, which generalizes the results on closed four-manifolds by M. Gursky and on conformally compact Einstein four-manifolds by S.-Y. A. Chang, J. Qing, and P. Yang. In the second part, we study Weyl functional on four-manifolds with boundary and establish several conformally invariant rigidity theorems. As applications, we prove some ri…
We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kahler-Einstein manifolds. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in odd dimensions. As a by-product, we obtain some Riemannian products with vanishing Q-curvature.
We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally related to a metric with vanishing Cotton tensor. One set of invariants we derive generalises the set of invariants in dimension four obtained by Kozameh, Newman and Tod. For the conformally Einstein problem, another set of invariants we construct gives necessary and sufficient conditi…
A new definition of canonical conformal differential operators $P_k$ ($k=1,2,...)$, with leading term a $k^{\rm th}$ power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham-Jenne-Mason-Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the $P_k$ operato…
We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This leads to an explicit description of $Q$-curvature and gauge companion operators on differential forms.
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincaré-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with geodesics away from the conformal singularity set and are shown to satisfy natural "boundary conditions" at points where they meet or cross the metric singularity set. We also characterise when they coincide with conformal circles. In the case of (Poincaré-)Einstein manifolds, we are able to provide a very general theory of first integrals for these distinguished…
We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-dimensional Einstein universe and we show that they can be integrated by quadratures in terms of elliptic functions, elliptic integrals and Jacobi's theta functions.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to -1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As consequences we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 2.
Poincaré-Einstein 4-manifolds with conformally Kähler geometry
本文研究共形类含Kähler度量的4维Poincaré-Einstein流形，将Einstein方程约化为Toda型方程，针对$\u212d^1$作用情形构建Dirichlet边值问题并证明存在唯一性，非微扰构造出共形无穷远为非正Yamabe型的无穷多族新度量。
We give a classification of compact conformally Kahler Einstein-Weyl manifolds whose Ricci tensor is hermitian.
Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.
Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.
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.
We propose further conformal parametrizations for initial data in some modified Einstein gravity theories. Some of them give rise to conformally covariant systems.
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of th…
We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ``twisted'' version of the recent generalisation of Hopf's rigidity result obtained by Burago and Ivanov.
In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.
In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$…
First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and quasi-Einstein ($η$-Einstein) conditions. Finally, we provide local classification and characterization of a semi-Riemannian (including the 4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic conformal (including closed) vector field, in terms of Petrov types and Bach tensor.
We present a global conformal invariant on closed six-manifolds which obstructs the existence of a conformally Einstein metric. We show that this obstruction is nontrivial and, up to multiplication by a constant, is the unique such invariant. This also gives rise to a (possibly trivial) diffeomorphism invariant which obstructs the existence of an Einstein metric. We also discuss global conformal invariants which obstruct the existence of a conformally Einstein metric on closed six-manifolds with infinite fundamental group.
We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity.
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.
Instability of conformally Kähler, Einstein metrics
本文证明了非半共形平坦、具有非负标量曲率的共形凯勒紧致或ALF爱因斯坦4-流形的不稳定性，并应用于所有已知非超凯勒引力瞬子及Chen-Lebrun-Weber度量。
We discuss a number of topics in the area of conformally compact Einstein metrics, mostly centered around the global existence question of finding such metrics with an arbitrarily prescribed conformal infinity. The paper is partly a survey of this area but also presents new results and a number of open problems.
There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L^\ell_k$ in terms of the null spaces of mutually commuting second…
Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein-Maxwell equations on the smooth 4-manifolds S^2 x S^2 and CP_2 # (-CP_2).
We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.
We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the hyperbolic plane with the real line are large-scale conformally rigid. This implies new characterizations of groups that can act properly, cocompactly by isometries on those manifolds.
A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The problem of local description of self-dual Einstein-like four-manifolds is also treated. A complete explicit solution of the Stäckel system in dimension four is obtained.
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee.
In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds.
In this paper, we present some new properties for p-biharmonic hypersurfaces in Riemannian manifold. We also characterize the p-biharmonic submanifolds in an Einstein space. We construct a new example of proper p-biharmonic hypersurfaces. We present some open problems.
Rigidity of Schouten Tensor under Conformal Deformation
本文研究了共形形变下 Schouten 张量有下界的度量刚性，利用该张量非负与 Ricci pinching 条件的等价性，为 Cheng 关于局部共形平坦流形的紧致性定理提供了简单证明。
This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.
The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…
In this note we prove that a special family of Killing potentials on certain Hirzebruch complex surfaces, found by Futaki and Ono, gives rise to new conformally Kähler, Einstein-Maxwell metrics. The correspondent Kähler metrics are ambitoric but they are not given by the Calabi ansatz. This answers in positive questions raised by Futaki-Ono.
Peng Wu recently announced a beautiful characterization of conformally Kaehler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det (W^+) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in (LeBrun 2015).
The main result of this paper is that the space of conformally compact Einstein metrics on a given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4, and a local existence and uniqueness theorem for such metrics with prescribed metric and stress-energy tensor at conformal infinity, again in dimension 4. This result also holds for Lorentzian-Einstein metrics with a positive cosmological constant.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.
In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^3, [\hat{g}])$ is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is a Berger metr…
We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge Kähler class a conformally Kähler, Einstein--Maxwell metric, or more generally, a Kähler metric of constant $(ξ, a, p)$-scalar curvature, then this metric minimizes the $(ξ,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson and developed by Li and Sano--Tipler, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--Tønnesen-Friedman, we describe the Kähler classes on a geometr…
We obtain a structure theorem for the group of holomorphic automorphisms of a conformally Kähler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the Kähler-Einstein, cscK, and extremal Kähler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally Kähler, Einstein--Maxwell metrics on $\mathbb{CP}^1 \times \mathbb{CP}^1$. We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of $(K, q, a)$-extremal Kähler metrics in a …
Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.
We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.
In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly Kähler compact manifolds.
We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$ and also the rigidity result when certain renormalized volume is zero.
On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III
本文研究了维度 $d\ge 4$ 的符合紧致爱因斯坦度量的紧致性及全局唯一性，并探讨了共形不变量的间隙现象。
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in \cite{GL} \cite{Lee} \cite{Bi} of conformally compact Einstein metrics with conform…
In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.
We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.
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