Hyperideal polyhedra in the 3-dimensional anti-de Sitter space

in Advances in Mathematics (2022), 404(Paper No. 108441, 61 pp),

We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all ... [more ▼]

We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all outside of AdS3 and whose edges all meet AdS3. We show that hyperideal polyhedra in AdS3 are uniquely determined by their combinatorics and dihedral angles, as well as by the induced metric on their boundary together with an additional combinatorial data, and describe the possible dihedral angles and the possible induced metrics on the boundary. [less ▲]

No Ensemble Averaging Below the Black Hole Threshold

in Journal of High Energy Physics (2022), 7

In the AdS/CFT correspondence, amplitudes associated to connected bulk manifolds with disconnected boundaries have presented a longstanding mystery. A possible interpretation is that they reflect the ... [more ▼]

In the AdS/CFT correspondence, amplitudes associated to connected bulk manifolds with disconnected boundaries have presented a longstanding mystery. A possible interpretation is that they reflect the effects of averaging over an ensemble of boundary theories. But in examples in dimension D≥3, an appropriate ensemble of boundary theories does not exist. Here we sharpen the puzzle by identifying a class of "sub-threshold" observables that we claim do not show effects of ensemble averaging. These are amplitudes that do not involve black hole states. To support our claim, we explore the example of D=3, and show that connected solutions of Einstein's equations with disconnected boundary never contribute to sub-threshold observables. To demonstrate this requires some novel results about the renormalized volume of a hyperbolic three-manifold, which we prove using modern methods in hyperbolic geometry. Why then do any observables show apparent ensemble averaging? We propose that this reflects the chaotic nature of black hole physics and the fact that the Hilbert space describing a black hole does not have a large N limit. [less ▲]

On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces

E-print/Working paper (2021)

The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K≥0 on the sphere is induced on the boundary of a unique convex body in $\R^3$. The ... [more ▼]

The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature K≥0 on the sphere is induced on the boundary of a unique convex body in $\R^3$. The answer was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a ``dual'' statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later proved. We describe three conjectural generalizations of the Weyl problem in $\HH^3$ and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry since a number of recent results and well-known open problems can be considered as special cases. One focus is on convex domain having a ``thin'' asymptotic boundary, for instance a quasicircle -- this part of the problem is strongly related to the theory of Kleinian groups. A second direction is towards convex subsets with a ``thick'' ideal boundary, for instance a disjoint union of disks -- here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is towards complete, convex disks of infinite area in $\HH^3$ and surfaces in hyperbolic ends -- with connections to questions on circle packings or grafting on the hyperbolic disk. Similar statements are proposed in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Half-pipe geometry. We also collect some partial new results mostly based on recent works. [less ▲]

A hyperbolic proof of Pascal's Theorem

in Mathematical Intelligencer (2021), 43(2), 130--133

We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Möbius, using hyperbolic geometry.

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

in Geometry and Topology (2021), 25-6

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a ... [more ▼]

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature K∈[−1,0) and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes. [less ▲]

Polyhedra inscribed in a quadric

in Inventiones Mathematicae (2020), 221(1), 237-300

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main ... [more ▼]

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the sphere and $\Gamma$ admits a Hamiltonian cycle. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout. [less ▲]

Constant Gauss curvature foliations of AdS spacetimes with particles

in Transactions of the American Mathematical Society (2020), 373(6), 4013--4049

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time N with particles (cone singularities of angles less than π along time-like curves), the ... [more ▼]

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time N with particles (cone singularities of angles less than π along time-like curves), the complement of the convex core in N admits a unique foliation by constant Gauss curvature surfaces. This extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe a parametrization of the space of convex GHM AdS metrics on a given manifold, with particles of given angles, by the product of two copies of the Teichm\"uller space of hyperbolic metrics with cone singularities of fixed angles. Finally, we use the results on K-surfaces to extend to hyperbolic surfaces with cone singularities of angles less than π a number of results concerning landslides, which are smoother analogs of earthquakes sharing some of their key properties. [less ▲]

Hyperbolic ends with particles and grafting on singular surfaces

in Annales de L'Institut Henri Poincaré. Analyse Non Linéaire (2019), 36(1), 181-216

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality ... [more ▼]

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichm\"uller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by "smooth grafting". [less ▲]

Properness for circle packings and Delaunay circle patterns on complex projective structures

E-print/Working paper (2018)

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex ... [more ▼]

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper. [less ▲]

The renormalized volume and uniformisation of conformal structures

in Journal de l'institut de mathématiques de Jussieu (2018), 17(4), 853-912