This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability in this class depends on the topology of the filling manifold. The obstruction to solvability for arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.
We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.
We prove that an $(n+1)$-dimensional spin static vacuum with negative cosmological constant whose null infinity has a boundary admitting a non-trivial Killing spinor field is the AdS spacetime. As a consequence, we generalize previous uniqueness results by X. Wang \cite{Wa2} and by Chru{ś}ciel-Herzlich \cite{CH} and introduce, for this class of spin static vacua, some Lorentzian manifolds which are prohibited as null infinities.
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.
We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.
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.
Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the immersion $S^2\to\mathbb{R}^3$.
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $γ= 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.
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.
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.
We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension $4$, and an existence theorem which holds in dimensions $n \geq 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the $\rm{U}(2)$-invariant Leray-Schauder degree for a family of negative-mass orbifolds found by LeBrun.
We generalize Penrose's notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Lifshitz and Schrodinger spaces (proposed as AdS/CFT duals of nonrelativistic field theories), warped AdS_3, and the near-horizon extreme Kerr geometry. The anisotropic conformal boundary appears crucial for resolving puzzles of holographic renormalization in such spacetimes.
In this paper we prescribe a fourth order conformal invariant on the standard $n-$sphere, with $n\geq5$, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-perturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.
We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed sequence converges. This implies that there exists a smooth conformal mapping, which minimizes the Willmore functional in this class. For this purpose we extend the quaternionic function theory of Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the Plu…
In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $σ_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $σ_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\ma…
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 give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive consta…
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 $α\in [0, 2]$.
Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$γ$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by --n and its scalar curvature approaches --n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M, [$γ$]) $\ge$ 0, where Y($\partial$M, [$γ$]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by J. …
This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results …
In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-invariant metric on $S^{2k+1}$ for $k\geq1$, the non-positively curved CCE metric on the $(2k+1)$-ball $B_1(0)$ with $(S^{2k+1}, [\hat{g}])$ as its conformal infinity is unique up to isometries. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of th…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum $(M^n,g,V),$ we also compute the asymptotic expansions of $g$ and $V$ at conformal infinity.
Static vacuum metrics in (4 + 1) dimensions with Λ < 0 and squashed conformal infinity
School of Physics, Nankai University（南开大学物理学院）；Mathematical Sciences and STAG Research Centre, University of Southampton（南安普顿大学数学科学与STAG研究中心）；TU Wien, Institut für Analysis und Scientific Computing（维也纳工业大学分析与科学计算研究所）；Department of Mathematics, KTH Royal Institute of Technology（瑞典皇家理工学院数学系）；Beijing Institute of Mathematical Sciences and Applications（北京应用数学与交叉科学研究院）；Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心）
It is well known that the spacetime $\text{AdS}_2\times S^2$ arises as the `near horizon' geometry of the extremal Reisser-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. Motivated by a conjectural viewpoint of Juan Maldacena, the authors in [4] studied the rigidity of asymptotically $\text{AdS}_2\times S^2$ spacetimes satisfying the null energy condition. In this paper, we take an entirely different and more general approach to the asymptotics based on the notion of conformal infinity. This involves a natural modification of the usual…
We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.
In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives estimates on the Morse index of the solutions and has the advantage to extend known existence results. Moreover it provides, for generic $K$ {\it Morse Inequalities at Infinity}, which give a lower bound on the numb…
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $\partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In…
Let $(M,g)$ be a compact Riemannian manifold on dimension $n \geq 4$ not conformally diffeomorphic to the sphere $S^n$. We prove that a smooth function $f$ on $M$ is a critical function for a metric $\tilde{g}$ conformal to $g$ if and only if there exists $x \in M$ such that $f(x)>0$.
We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.
We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-Δ)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the vo…
We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on $M$.
In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.
The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact R…
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.
In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain…
The paper is an attempt to resolve the prescribed Chern scalar curvature problem. We look for solutions within the conformal class of a fixed Hermitian metric. We divide the problem in three cases, according to the sign of the Gauduchon degree, that we analyse separately. In the case where the Gauduchon degree is negative, we prove that every non-identically zero and non-positive function is the Chern scalar curvature of a unique metric conformal to the fixed one. Moreover, if there exists a balanced metric with zero Chern scalar curvature, we prove that every smooth function changing sign wit…
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…
For odd dimensional Poincaré-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for small $m$ but it becomes finite dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\bar{X},\pl\bar{X})$ and the kernel of the Branson-Gover \cite{BG} differential operators $(L_k,G_k)$ on the conformal infinity $(\pl\bar{X},[h_0])$. In a second time we relate the set of $C^{n-2k+1}(Λ^k(\bar{X…
We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.
In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the conformal invariants of the conformal infinity and its ambient.
We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix is shown to be a pseudodifferential operator. For generic frequency in this part of the continuous spectrum, we give an explicit construction of the resolvent kernel.
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.
