Rafe Mazzeo, Frank Pacard
We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics; this requires no nondegeneracy hypothesis. We also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join.
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For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
A Poincaré-Einstein metric $g$ is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie in $L^2$. We prove that a 4-dimensional Poincaré-Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $R_+$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $R_+$ is negative definite then $g$ is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré-Einstein metric of negative sectional curvature is non-degenerate
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.
In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball $B^8$. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a suitable compactification).
Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.
Self-dual and even Poincaré-Einstein metrics in dimension four
该研究证明四维自对偶偶Poincaré-Einstein度量的刚性与间隙定理，借边界共形不变量和体拓扑给出其存在阻碍，还发现三维黎曼流形的新标量共形不变量。
We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincaré-Einstein manifold of one lower dimension with an isolated conical singularity.
We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.
We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,α}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its …
We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincare-Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classifica…
On the Poincaré-Einstein manifolds with cylindrical conformal infinity
本文证明具有柱形共形无穷远的非正曲率Poincaré-Einstein流形必为双曲，并在圆因子长度足够大或Weyl曲率能量足够小时给出唯一性，通过建立新的ε-正则性实现。
We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.
Let $(M,g)$ be a compact Kähler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré--Einstein metrics on $X$. One 1-parameter subfamily $\{g_t\}_{t>0}$ has the property that as $t\searrow 0$, $g_t$ converges to a PE metric $g_0$ on $X_0$ with conic singularity, while $t^{-1}g_t$ converges to a complete Ricci-flat Kähler metric $\hat{g}_0$ on $X$. Another 1-param…
Ambient and Poincaré metrics for CR $3$-manifolds associated with the self-dual Einstein ACH metric
本文从自对偶爱因斯坦ACH度量出发，为3维CR流形的Fefferman共形流形构造了新的环境与庞加莱度量，其满足无穷阶爱因斯坦-麦克斯韦型方程，可用于构造CR不变算子，还实现了3维所有阶CR GJMS算子的环境度量构造。
Dynamical Stability and Instability of Poincaré--Einstein Manifolds
本文利用渐近双曲流形的扩张子熵变体，证明了Ricci流下Poincaré-Einstein度量的动力学稳定性与不稳定性，并通过局部正质量定理和体积比较进行刻画。
We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential operators. We study the scattering matrix by using the 0-calculus and generalize a result of Graham for the case of the standard hyperbolic metric on a ball.
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 …
Let X be a Kähler manifold and D be a R-divisor with simple normal crossing support and coefficients between 1/2 and 1. Assuming that K_X+D is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X\D having mixed Poincaré and cone singularities according to the coefficients of D. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair (X,D).
Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincare metrics we construc…
We provide the first example of continuous families of Poincaré-Einstein metrics developing cusps on the trivial topology $\mathbb{R}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański-Demiański. We additionally indicate how to construct similar examples on more complicated topologies.
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。
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.
An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincaré--Einstein metric. Such conformal rescalings are in general obstructed by conformal invariants of the boundary hypersurface embedding, the first of which is the trace-free second fundamental form and then, at the next order, the trace-free Fialkow tensor. We show that these tensors are the lowest order examples in a sequence of conformally invariant higher fundamental forms determined by the data of a conformal hypersurface embedding. We give a construction of the…
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined PDE on a function. Away from the zeros of this the almost Einstein structure is Einstein, while the zero set gives a scale singularity set which may be viewed as a conformal infinity for the Einstein metric. In this article we give a classification of the possible scale singular…
We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.
On the problem of filling by a Poincaré-Einstein metric in dimension 4
本文研究四维空间中通过庞加莱-爱因斯坦度量填充问题，建立紧致性结果并推导填充存在性结论。
A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.
Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and $Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In this paper, we first show that …
Poincaré-Einstein 4-manifolds with cusps
本文构造了多种带尖点的庞加莱-爱因斯坦4流形，包括无限族爱因斯坦度量，并回答了Anderson关于尖点形成的问题。
Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincaré-Einstein Manifolds
本文为Poincaré-Einstein流形引入共形协变边界算子，建立高阶Dirichlet问题以恢复分数阶GJMS算子，并推导出相关高阶迹Sobolev不等式，最终结合Beckner不等式得到球上的尖锐高阶迹不等式。
In this paper, we show that for a Poincaré-Einstein manifold $(X^{n+1},g_+)$ with conformal infinity $(M,[\hat{g}])$ of nonnegative Yamabe type, the fractional Yamabe constants of the boundary provide lower bounds for the relative volume. More explicitly, for any $γ\in (0,1)$, $$ \left(\frac{Y_{2γ} (M,[\hat{g}])}{Y_{2γ} (\mathbb{S}^n , [g_{\mathbb{S}}])}\right)^{\frac{n}{2γ}} \leq \frac{V(Γ_t(p),g_+)}{V(Γ_t(0),g_{\mathbb{H}})} \leq \frac{V( B_t(p),g_+)}{V( B_t(0), g_{\mathbb{H}})} \leq 1,\quad 0<t<\infty, $$ where $B_t(p)$, $Γ_t(p)$ are the the geodesic ball and geodesic sphere of radius $t$ i…
The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings
本文研究 Poincaré-Einstein 填充的 Dirichlet-to-Neumann 映射，构造共形不变的超曲面不变量，并证明其在偶数维情形下的唯一性，同时推广到奇数维共形平坦边界情形。
Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness
针对球面三维空间形式M=S³/Γ，通过四维庞加莱-爱因斯坦填充结合等变Fredholm奇偶性，证明其Yamabe常数的Sharp界为|Γ|^(-2/3)Y(S³)，得到了该类空间形式的Yamabe常数精确值。
Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds
针对局部平坦的Poincaré-Einstein流形共形无穷远处的分数次Yamabe问题，用Bahri-Coron代数拓扑方法规避正质量定理缺失问题，证明存在常分数次数量曲率黎曼度量。
The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian $n$-manifold $(M,g)$ and a positive function $N$, called the lapse. We study this problem on Asymptotically Poincaré-Einstein $n$-manifolds, $n\ge 3$, when the conformal boundary-at-infinity is either a round sphere, a flat torus or smooth quotient thereof, or a compact hyperbolic manifold. Such manifolds have well-defined Wang mass, and are time-symmetric slices of st…
We examine here the space of conformally compact metrics $g$ on the interior of a compact manifold with boundary which have the property that the $k^{th}$ elementary symmetric function of the Schouten tensor $A_g$ is constant. When $k=1$ this is equivalent to the familiar Yamabe problem, and the corresponding metrics are complete with constant negative scalar curvature. We show for every $k$ that the deformation theory for this problem is unobstructed, so in particular the set of conformal classes containing a solution of any one of these equations is open in the space of all conformal classes…
We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter al…
For a class of even dimensional conformally compact manifolds (X,g), we define a generalized Krein spectral function by applying a renormalized trace functional to the spectral measure of the Laplacian. We then show that this is the phase of the Kontsevich-Vishik determinant det S(s) of the scattering operator S(s) of (the Laplacian of) g and we analyze the divisors of det(S(s)). As an application for convex co-compact hyperbolic quotients, we obtain a functional equation for Selberg's Zeta function Z(s), we express the determinant of the GJMS conformal Laplacians of the conformal infinity of …
Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds
本文研究庞加莱-爱因斯坦流形中任意余维极小子流形的重整化体积，推导其一阶和二阶变分公式，并给出边界柱面上的渐近描述及双曲空间情形下的内积关系。
This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.
Nahm Poles and 0-Instantons
研究渐近超曲4流形上的自对偶0连通，或0-瞬子，其在共形无穷处具有均匀奇异性，并渐近趋于H^4上的Nahm极点模型解。证明了展开是log-smooth的，且首log项系数（0-瞬子障碍张量）是与周围共形度量的Weyl曲率相关的共形不变量。
Holography and Cheeger constant of asymptotically CMC submanifolds
本文研究渐近双曲流形中渐近 CMC 子流形的 Cheeger 常数，并刻画弱 Poincaré--Einstein 条件及渐近极小超曲面平均曲率二阶消失的共形障碍。
The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,g)$ must have mass not less than the mass $m_0$ of a metric $g_0$ induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics …
Variation of Kahler-Einstein metrics with mixed singularities
本文研究了具有混合奇点的Kähler-Einstein度量的变化，推广了Schumacher和Guenancia的结果，并证明了光滑log canonical对的Albanese映射的满射性。
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functional on the manifold.
Conformal hypersurface invariants and Bach-type Boundary Problems
本文利用变分方法发现偶数维共形流形中一个新的对称无迹张量超曲面共形不变量，证明其完善了共形基本形式族，并据此解决了Poincaré--Einstein问题中的Dirichlet--to--Neumann映射像问题，同时构造了边界的新全局不变量及Bach平坦流形的度量。
Einstein-Yang-Mills fields in conformally compact manifolds
研究共形紧渐近局部双曲流形上Einstein-Yang-Mills场的形变理论，利用$0$-微积分证明平凡且非退化的场在无穷远边界数据小扰动下可实现为Einstein-Yang-Mills场，推广了先前结果。
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