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 curves. As well as the general procedure outlined, a specific example is given.
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.
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 derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…
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…
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than $\frac {n-α}2$ for some $α\in [2, n)$, the set of positive smooth solutions to the equation $$ P_αu = u^\frac {n+α}{n-α} $$ is compact in the $C^\infty$ topology. There…
We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an …
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.
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only if the conformal infinity of $(X,g)$ is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian $P(α)$ on the conformal infinity is non-negative for all $α\in [0, 2]$.
In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.
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 $[g_M]$.
In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^\infty$ topology.
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 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.
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 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…
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with distinguished curves in a Lagrangian submanifold of a symplectic manifold. For this purpose, we will combine the concept of a Tulczyjew triple with Marsden-Weinstein symplectic reduction.
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
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 …
In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2γ}$ for the conformal infinity. If a Poincaré-Einstein manifolds $(X^{n+1}, g_+)$ is locally conformally flat and there exists an representative $g$ for the conformal infinity $(M, [g])$ such that the scalar curvature $R$ is a positive constant and $Q_4>0$, then we prove that $P_{2γ}$ is positive for $γ\in (1,2)$ and thus the first real scattering pole is less than $\frac{n}{2}-2$.
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 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…
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.
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifolds. Our boundary operators also provide a new realization of the fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev trace inequalities involving the interior $W^{3,2}$-seminorm, includ…
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 …
We show that the Teukolsky connection, which defines generalized wave operators governing the behavior of massless fields on Einstein spacetimes of Petrov type D, has its origin in a distinguished conformally and GHP covariant connection on the conformal structure of the spacetime. The conformal class has a (metric compatible) integrable almost-complex structure under which the Einstein space becomes a complex (Hermitian) manifold. There is a unique compatible Weyl connection for the conformal structure, and it leads to the construction of a conformally covariant GHP formalism and a generaliza…
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.
Let $X$ be an asymptotically hyperbolic manifold and $M$ its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on $M$ under various geometric assumptions on $X$ and $M$: Firstly, we handle when the boundary $M$ has a point at which the mean curvature is negative. Secondly, we re-encounter the case when $M$ has zero mean curvature and is either non-umbilic or umbilic but non-locally conformally flat. As a result, we replace the geometric restrictions given by González-Qing (Analysis and PDE, 2013) and González-Wang (arXiv:1503.02862) …
A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting globa…
We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the $σ_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the $σ_k$-Ricci problem. By adopting results of Mazzeo-Pacard, we show an interesting relationship between the complete metrics we construct and the existence of Poincaré-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case…
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, the geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is …
We argue that the AdS/CFT calculational prescription for double-trace deformations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a quantum 1-loop correction to the SUGRA action and the boundary counterpart accounts for a sub-leading term in the large-N limit. The sequence of GJMS conformal Laplacians shows up in the two-point function of the CFT operator dual to a bulk scalar field at certain values of its …
We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For th…
Conformal qc geometry of spherical qc manifolds are investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. On a scalar positive spherical qc manifold, we can construct the Green function of the qc Yamabe operator, which can be applied to construct a conformally invariant tensor. It becomes a spherical qc metric if the qc positive mass conjecture is true. Conformal qc geometry of spher…
