Recent publications (since 2020)
Here are some recent publications (since 2020).
Preprints not yet published (or accepted) are here,
papers published before 2009 are there.
There are some other texts (surveys, etc) on
another page.
Please send me an e-mail (or a s-mail) if you wish to receive a preprint
or a reprint.
If you have a Mathscinet access you can check the
Math Reviews of
those papers and of some older stuff.
No Ensemble Averaging Below the Black Hole Threshold.
Jean-Marc Schlenker, Edward Witten.
arXiv:2202.01372.
J. High Energy Physics, JHEP 7(2022)143 (25th anniversary issue).
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 ''fixed energy''
or ''sub-threshold'' observables that we claim do not show effects of ensemble averaging.
These are amplitudes that involve states that are above the ground state by only a fixed
amount in the large N limit, and in particular are far from being 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 these 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.
Hyperideal polyhedra in the 3-dimensional anti-de Sitter space.
Qiyu Chen and Jean-Marc Schlenker.
arXiv:1904.09592.
Adv. Math. 404B, 2022, 108441.
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.
Bending laminations on convex hulls of anti-de Sitter quasicircles.
Louis Merlin, Jean-Marc Schlenker.
arXiv:2006.13470.
Proc. London Math. Soc. 123(2021):4, 410--432.
Let $\lambda_-$ and $\lambda_+$ be two bounded measured laminations on the
hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below).
We consider the left earthquakes along $\lambda_-$ and $\lambda_+$,
considered as maps from the universal Teichm\"uller space $\mathcal T$ to
itself, and we prove that the composition of those left earthquakes has a fixed
point.
The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism
$u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in
$AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial
AdS^3$ is the disjoint union of two embedded copies of the hyperbolic plane,
pleated along measured geodesic laminations. Our main result is that any pair
of bounded measured laminations that "strongly fill" can be obtained in this
manner.
A hyperbolic proof of Pascal's Theorem.
Miguel Acosta and Jean-Marc Schlenker.
arXiv:2012.14883.
The Mathematical Intelligencer, 43(2021):2, 130--133.
We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as
a generalization by M\"obius, using hyperbolic geometry.
NB: general-audience expository paper.
Weakly Inscribed Polyhedra.
Hao Chen and Jean-Marc Schlenker.
arXiv:1709.10389.
Trans. Amer. Math. Soc., Series B 9(2022) 415--449.
We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices
on a sphere. We do not require, in particular, that the polyhedra lie in the
interior of the sphere, hence the term "weakly inscribed". Such polyhedra can
be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a
combination of the hyperbolic space and the de Sitter space, with the sphere as
the common ideal boundary. We have three main results: (1) the $1$-skeleta of
weakly inscribed polyhedra are characterized in a purely combinatorial way, (2)
the exterior dihedral angles are characterized by linear programming, and (3)
we also describe the hyperbolic-de Sitter structure induced on the boundary of
weakly inscribed polyhedra.
Quasicircles and width of Jordan curves in CP1.
Francesco Bonsante, Jeffrey Danciger, Sara Maloni, Jean-Marc Schlenker.
arXiv:1908.09175.
Bull. of the London Math. Soc., 53(2021):2, 507--523.
We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles.
The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry.
Francesco Bonsante, Jeffrey Danciger, Sara Maloni, Jean-Marc Schlenker.
arXiv:1902.04027.
Geometry & Topology 25 (2021) 2827-2911.
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\in [-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.
Volumes of quasifuchsian manifolds.
Jean-Marc Schlenker.
arXiv:1903.09849.
Surveys in Differential Geometry, 25:1(2020), 319-353.
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact
hyperbolic manifolds, have infinite volume, but they have a well-defined
``renormalized'' volume. We outline some relations between this renormalized
volume and the volume, or more precisely the ``dual volume'', of the convex
core. On one hand, there are striking similarities between them, for instance
in their variational formulas. On the other, object related to them tend to be
within bounded distance. Those analogies and proximities lead to several
questions. Both the renormalized volume and the dual volume can be used for
instance to bound the volume of the convex core in terms of the Weil-Petersson
distance between the conformal metrics at infinity.
Flipping Geometric Triangulations on Hyperbolic Surfaces.
Vincent Despré, Jean-Marc Schlenker, Monique Teillaud.
36th International Symposium on Computational Geometry (SoCG 2020), 35:1--35:16.
arXiv:1912.04640.
We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation.
Polyhedra inscribed in a quadric.
Jeffrey Danciger, Sara Maloni, Jean-Marc Schlenker.
arXiv:1410.3774.
Inventiones Mathematicae, 221(2020):1, 237-300. Online version.
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 G is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if G is realized as the 1-skeleton of a polyhedron inscribed in the sphere and ¿ 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.
Constant Gauss curvature foliations of AdS spacetimes with particles.
Qiyu Chen, Jean-Marc Schlenker.
arXiv:1610.07852.
Transactions of the American Math. Soc., 373(2020):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 $\pi$ 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 Teichmüller 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 $\pi$ a number of results concerning landslides, which are smoother analogs of earthquakes sharing some of their key properties.
Older papers
