In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $d\ge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $d\ge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.
An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham expansions. Moreover we show that conformal classes of metrics that are near that of the round metric on the n-sphere have fillings into the ball satisfying the Lovelock equation, extending the existence result of Graham-Lee for Einstein metrics.
We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.
We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, or a single point. In the latter case, we show that $S$ must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham cohomology classes of Lee forms of locally conformally Kähler structures of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, …
Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realized in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realization we show in addition t…
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.
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.
This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper addresses the hardest challenge in this series: It shows how to {\it separate} the local conformal invariant from the divergence term in the integrand; …
A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.
We study the set of deRham classes of Lee $1$-forms of the locally conformally symplectic (LCS) structures taming the complex structure of a compact complex surface in the Kodaira class VII, and show that the existence of non-trivial upper/lower bounds with respect to the degree function correspond respectively to the existence of certain negative/non-negative PSH functions on the universal cover. We use this to prove that the set of Lee deRham classes of taming LCS is connected, as well as to obtain an explicit negative upper bound for this set on the hyperbolic Kato surfaces. This leads to a…
Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,θ)$ a $d$(bounded) locally conformally Kähler manifold if the lifted Lee form $\tildeθ$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}χ(M^{2n})\geq$ (resp. $>$) $0$.
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.
We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operators is used to describe the second variation of the total Q-curvature. An explicit formula for conformally Einstein manifolds is given in terms of the Lichnerowicz Laplacian of an Einstein representative metric.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conformally covariant pseudodifferential operator. In the setting of Poincaré--Einstein manifolds, we show that these operators agree with the fractional GJMS operators of Graham and Zworski. We also use our operators to establish some new sharp Sobolev trace inequalities.
In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum within its conformal class, unless the manifold is isometric to round sphere with the standard metric up to a multiple of constant. Note that when $(M,g)$ is locally conformally flat, this functional reduces to the well-studied $\int_M σ_3(g)dv_g$. Hence, our result generalize a pr…
We study 4-dimensional second-Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe 4-dimensional compact second-Chern-Einstein locally conformally symplectic manifolds and we give some examples of such manifolds. Finally, we study the second-Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second-Chern-Einstein metric with a parallel non-zero Lee form.
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…
For a conformally compact Poincaré-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=ρ^2g_+$, where $ρ$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{g}_s=ρ^2_sg_+$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré-Einstein manifolds $\{(X, g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{\bar{g}^{(i)}\}$ is compact in $C^{k,α}(\overline{X})$ topology if and only if $\{\bar{g}_s^{(i)}\}$ is …
We define a conformally invariant action S on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler-Lagrange equations of S, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang-Mills equations. For any gauge connection A on M, we define S(A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q-curvature. Integrating this density provides t…
The notion of a Douglas space of second kind of a Finsler space with $(α, β)$-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space of second kind with special $% (α, β)$-metric $α+εβ+ k \frac{β^2}{α}$ is conformally transformed to a Douglas space of second kind. Further, we obtain some results which prove that a Douglas space of second kind with certain $(α, β)$-metrics such as Randers metric, Kropina metric, first approximate Matsumoto metric and Finsler space with square metric is conformally t…
We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperb…
A locally conformally Kahler manifold is a Hermitian manifold $(M,I,ω)$ satisfying $dω=θ\wedge ω$, where $θ$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nablaθ$ is of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and Vaisman if $\nablaθ=0$. We show that a compact LCK manifold is pluricanonical if and only if the Lee form has constant length and the Kahler form of its covering admits an automorphic potential. Using a degenerate Monge-Ampere equation and the classification of surfaces of Kahler rank one, due to Brunella, Chiose and T…
Morse-Novikov cohomology of locally conformally Kähler manifolds
局部共形Kahler（LCK）流形是一个具有Kahler覆盖的复流形，其中单值映射通过同素映射作用于该覆盖上。我们定义了三个上同调不变量，李类，Morse-Novikov类和Bott-Chern类，LCK结构。这些不变量一起扮演着与Kahler几何中的Kahler类相同的角色。如果两个LCK结构的这些类重合，则这些结构之间的差异可以由光滑势表示，类似于Kahler的情况。我们证明了Vaisman流形的Morse-Novikov类和Bott-Chern类为零。此外，对于Vaisman流形上的任何LCK-结构，我们证明了它的Morse-Novikov类为零。证明了Bott-Chern类为零的紧致LCK-流形$M$在Hopf流形上有一个全纯嵌入，如果$\dim_\C M \geq 3$，这一结果与Kodaira嵌入定理相似.
We generalize the splitting theorem of Cai-Galloway for complete Riemannian manifolds with $\Ric\geq-(n-1)$ admitting a family of compact hypersurfaces tending to infinity with mean curvatures tending to $n-1$ sufficiently fast to the setting of smooth metric measure spaces. This result complements and provides a new perspective on the splitting theorems recently proven by Munteanu-Wang and Su-Zhang. We show that the mean curvature assumption in our result is sharp, which also provides an example showing that the assumption $R\geq-(n-1)$ in the Munteanu-Wang splitting theorem for expanding gra…
It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant $Q$-curvat…
Oeljeklaus-Toma (OT) manifolds are complex non-Kähler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT …
