We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of our surfaces $(M,g)$ cannot collapse as the conformal structure degenerates which is important in applications in which $(M,g)$ serves as domain of a PDE with boundary conditions.
We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.
In this short note, we show that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is also showed that the flow converges to a flat metric as time going to infinity.
We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.
Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
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.
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.
We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.
In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationally invariant regions and obtain the possible isoperimetric solutions by studying the behaviour of the profile curves of the rotational surfaces with constant mean curvature in the hyperbolic space.
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.
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…
This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.
We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The proof uses the total scalar curvature functional on the space of ``cusps with particles'', which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced m…
In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order near boundary have conformal compactifications that are $C^{m+2,α}$ up to the boundary when Weyl curvature is in $C^{m,α}$ and the boundary metric is in $C^{m+2,α}$ where $m\geq 3.$
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 positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.
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.
For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.
This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long time behavior of the flow and the geometric meaning is investigated.
We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete $1/4$-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics $g^c$, $c\ge 0$, on $\mathbb R^4$. The metric $g^0$ is the complex hyperbolic metric whereas the family $(g^c)_{c>0}$ is equivalent to a family of metrics $(h^b)_{b>0}$ depending on $b=1/c$ and smoothly extending to $b=0$ for which $h^0$ is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine th…
The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyperbolic spaces as our main result.
In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D'Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.
In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball $\mathbf{D}^3$, which contains multiple horizons.
We consider globally hyperbolic maximal anti de Sitter 3-manifolds $M$ with a closed Cauchy surface $S$ of genus greater than one and prove that any pair of hyperbolic metrics on $S$ can be realized as the boundary metrics of the convex core of a maximal globally hyperbolic anti de Sitter 3-manifold structure on $M$. This answers the existence part of a question of Mess about the unique realization of such metrics. Our theorem has a nice formulation purely in terms of 2-dimensional Teichmüller theory and earthquakes.
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian curvature measure.
We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere.
We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.
