Xiaoxiang Chai
We discuss asymptotically hyperbolic manifold with a noncompact boundary which is close to a horosphere in a certain sense. The model case is a horoball or the complement of a horoball in standard hyperbolic space. We show some geometric formulas.
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The classical Brunn-Minkowski theory studies the geometry of convex bodies in Euclidean space by use of the Minkowski sum. It originated from H. Brunn's thesis in 1887 and H. Minkowski's paper in 1903. Because there is no universally acknowledged definition of the sum of two sets in hyperbolic space, there has been no Brunn-Minkowski theory in hyperbolic space since 1903. In this paper, for any $p>0$ we introduce a sum of two sets in hyperbolic space, and we call it the hyperbolic $p$-sum. Then we develop a Brunn-Minkowski theory in the hyperbolic space by use of our hyperbolic $p$-sum, and we…
We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the harmonic analysis on symmetric spaces (Riemannian as well as pseudo-Riemannian ones) is equivalent (homologous), up to the Abelian Fourier transform, to the similar problem in the flat model. On the technical level it is important that we work not with the usual horospherical tra…
The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space
本文研究双曲空间中凸超曲面的对应几何问题，通过建立新满秩定理证明极限球面p-Christoffel-Minkowski问题存在光滑原点对称严格凸解，并提出给定p-偏移Weingarten曲率问题且证明其存在性结果。
The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space
本文研究双曲空间中的horospherical $p$-Christoffel-Minkowski问题，通过粘性方法证明满秩定理，建立了一致$h$-凸解的存在性，并蕴含了共形几何中Nirenberg型问题解的存在性。
Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane
本文研究双曲平面各向同性 horospherical $p$-Minkowski 问题，对临界参数 $p=-7$ 两侧解的唯一性与非唯一性进行分类，并推广到 $q$-加权情形。
Horospherical mean curvature functions and D'Atri spaces
本文研究满足特定horospherical平均曲率函数性质的无共轭点单连通黎曼流形，证明附加温和曲率条件下秩一流形渐近调和、紧秩一情形为负曲率局部对称空间，且同条件下秩一D'Atri空间调和，推广了Druetta的相关结果。
The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\H^n$. In the second part, we esta…
Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.
Using the upper half space model, we evaluate a component of the hyperbolic mass functional evaluated on a special family of polyhedra extending a formula of Miao-Piubello.
We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.
We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $\mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.
For asymptotically hyperbolic manifolds of dimension $n$ with scalar curvature at least equal to $-n(n-1)$ the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.
We provide a Hilbert manifold structure {à} la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincar{é} and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.
When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, we show that an asymptotically hyperbolic initial data set with non-negative local energy density can be approximated by an initial data set with strictly positive local energy density and a simple structure at infinity, while changing the mass arbitrarily little. This is achieve…
We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.
In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resultin…
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…
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.
We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.
We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality. Our results require that the scalar curvature of the metric satisfies $R_{g}\geq -6$, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without $R_{g}\geq -6$ so that large centered coordinate spheres are not isope…
We prove positivity of energy for a class of asymptotically locally hyperbolic manifolds in dimensions $4\le n \le 7$. The result is established by first proving deformation-of-mass-aspect theorems in dimensions $n\ge 4$. Our positivity results extend to the case $n = 3$ when more stringent conditions are imposed.
Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds $X$ with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon-Nykodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horosp…
A family of non-radial solutions of the Yamabe equation, with reference the hyperbolic space, is constructed using power series. As a result, we obtain a family of asymptotically hyperbolic metrics, with spherical conformal infinity, with scalar curvature greater than -n(n - 1), but which are a priori not complete. Moreover, any vector of R^n+1 is performed by an energy-momentun vector of one suitable metric of this family. They can in particular provide counter-examples to the positive energy-momentum theorem when one removes the completeness assumption.
We prove positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain locally hyperbolic manifolds. As an application we derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chruściel and Simon on the validity of a Penrose-type inequality for black hole solutions carrying a higher genus horizon.
We solve the Jang equation with respect to asymptotically hyperbolic "hyperboloidal" initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.
We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n>2 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstan…
Sharp estimates for the Robin Laplacian under a perimeter constraint in hyperbolic space
本文研究了双曲空间中Robin拉普拉斯算子的特征值问题，证明了测地球在特定条件下最大化特征值，并解决了相关开放问题。
A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric. This results extends in higher dimension, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds…
The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Using their method, Rigger proved the same theorem for Riemannian manifolds being asymptotically equal to the (spatial) (Schwarzschild-)Anti-de Sitter solution. This was generalized to asymptotically hyperbolic manifolds by Neves-Tian, Chodosh, and the author at a later stage. In this work, we prove the reverse implication as the author already did in …
On Jang's equation and the Positive Mass Theorem for asymptotically hyperbolic initial data sets with dimensions above three and below eight
本文在维度4至7的渐近双曲初始数据上求解Jang方程，为非旋量证明正质量定理提供了新途径，并推广了Sak21在三维情形下的结果。
We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite -- an altered version of -- the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic co…
We prove a positive mass theorem for complete Kähler manifolds that are asymptotic to the complex hyperbolic space.
Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.
This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asympto…
On Asymptotically Locally Hyperbolic Metrics with Negative Mass
本研究构建了带指定表观视界与共形边界拓扑、质量可控的常数量曲率渐近局部双曲黎曼度量，得到了新的负质量广义相对论初始数据集解。
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic s…
We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenever a geometrically motivated system of elliptic equations admits a solution. The inequalities treated here relate mass, angular momentum, charge, and horizon area.
In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.
In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $σ_{2}$ curvature has lower bound $σ_{2}\geq\frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space $\mathbb{H}^{4}$ when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.
We show that the mass of an asymptotically hyperbolic manifold with a noncompact boundary can be evaluated via the Ricci tensor and the second fundamental form by using purely coordinates. The method is analog to Miao-Tam's approach to the asymptotically flat manifold.
A definition of the mass aspect function for weakly regular asymptotically hyperbolic manifolds
本文针对弱正则渐近双曲流形，提出了一种ADM风格的质量方面函数定义，统一了Wang和Chruściel-Herzlich的两种质量定义方法，并证明了其可通过Ricci张量计算及在无穷远图表变换下的良好协变性质。
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a…
The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [20…
We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.
We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions $n\geq 3$ an optimal Penrose inequality for certain graphs in hyperbolic space $\mathbb H^{n+1}$ whose boundary has constant mean curvature $n-1$.
We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.
We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.
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.
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.
In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an asymptotically flat manifold.
We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(ρ^5)$ in dimensions three and higher.
Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in this sense. We use the geodesic compactification by asymptotic geodesic rays to compactify M and appropriate curvature decay conditions to study the regularity of the conformal compactification. We also present an interesting example that shows our conclusion is nearly optimal for our assumptions.
Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds
本文研究渐近双曲图形流形正质量定理的稳定性，借鉴Huang、Lee和Sormani的方法，证明当总质量几乎为零时该类流形在内蕴平坦距离意义下接近双曲空间。
We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries.
In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, C. -Y. Lin used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension and C. Sormani and Lin proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass.
In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein manifolds with the same conformal infinity. The construction involves finding a sequence of approximate solutions to the Einstein equations and then perturbing them to exact ones.
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