JeanMarc Schlenker
Mathematics department,
University of Luxembourg
University of Luxembourg
Recent preprints

Convex surfaces with prescribed induced metrics in antide Sitter spacetimes.
Qiyu Chen, JeanMarc Schlenker.
arXiv:2405.01912
Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric of curvature $K<1$ on $S$, and let $h_0$ be a hyperbolic metric on $S$. We show that there exists a unique quasifuchsian AdS spacetime with left metric isotopic to $h_0$, containing a pastconvex Cauchy surface with induced metric isotopic to $h$. 
Convex cocompact hyperbolic manifolds are determined by their pleating lamination.
Bruno Dular, JeanMarc Schlenker.
arXiv:2403.10090
Convex cocompact 3dimensional hyperbolic manifolds are uniquely determined by the pleating measured lamination on the boundary of their convex core. 
GromovThurston manifolds and antide Sitter geometry.
Daniel Monclair, JeanMarc Schlenker, Nicolas Tholozan.
arXiv:2310.12003
We consider hyperbolic and antide Sitter (AdS) structures on M×(0,1), where M is a ddimensional GromovThurston manifold. If M has cone angles greater than 2π, we show that there exists a "quasifuchsian" (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to M. When M has cone angles less than 2π, there exists a hyperbolic end with boundary a concave pleated surface isometric to M. Moreover, in both cases, if M is a GromovThurston manifold with 2k pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension 2k−3. When d=3, the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to M×(0,1) contains a submanifold of dimension 2k−2, and extends up to a "Fuchsian" manifold, that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by~$\R$. We use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of $Ø(2d,2)/\U(d,1)$. The construction uses an explicit correspondence between quasifuchsian 2d+1dimensional AdS manifolds and compact quotients of $Ø(2d,2)/\U(d,1)$ which we interpret as the space of timelike geodesic Killing fields of $\AdS^{2d+1}$. 
The geometric data on the boundary of convex subsets of hyperbolic manifolds.
Qiyu Chen, JeanMarc Schlenker.
arXiv:2210.11782
Let $N$ be a geodesically convex subset in a convex cocompact hyperbolic manifold $M$ with incompressible boundary. We assume that each boundary component of $N$ is either a boundary component of $\partial_\infty M$, or a smooth, locally convex surface in $M$. We show that $N$ is uniquely determined by the boundary data defined by the conformal structure on the boundary components at infinity, and by either the induced metric or the third fundamental form on the boundary components which are locally convex surfaces. We also describe the possible boundary data. This provides an extension of both the hyperbolic Weyl problem and the AhlforsBers Theorem. Using this statement for quasifuchsian manifolds, we obtain existence results for similar questions for convex domains $\Omega\subset \HH^3$ which meets the boundary at infinity $\partial_{\infty}\HH^3$ either along a quasicircle or along a quasidisk. The boundary data then includes either the induced metric or the third fundamental form in $\HH^3$, but also an additional ``gluing'' data between different components of the boundary, either in $\HH^3$ or in $\partial_\infty\HH^3$. 
The Weyl problem for unbounded convex domains in H^{3}.
JeanMarc Schlenker.
arXiv:2106.02101
Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the conformal structure on the boundary of $K$ in the Poincaré model of $\HH^3$, together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. We do not consider here the uniqueness of a convex subset with given boundary data. 
On the Weyl problem for complete surfaces in the hyperbolic and antide Sitter spaces.
JeanMarc Schlenker.
arXiv:2012.14754
The classical Weyl problem (solved by Lewy, Alexandrov, Pogorelov, and others) asks whether any metric of curvature $K\geq 0$ 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 wellknown 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 antide Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur, and in Minkowski and Halfpipe geometry. We also collect some partial new results mostly based on recent works.
NB: this is mostly a survey, with only limited new results. Not designed to be submitted at this stage. 
The prestige and status of research fields within mathematics.
JeanMarc Schlenker.
arXiv:2008.13244.
While the ``hierarchy of science'' has been widely analysed, there is no corresponding study of the status of subfields within a given scientific field. We use bibliometric data to show that subfields of mathematics have a different ``standing'' within the mathematics community. Highly ranked departments tend to specialize in some subfields more than in others, and the same subfields are also overrepresented in the most selective mathematics journals or among recipients of top prizes. Moreover this status of subfields evolves markedly over the period of observation (19842016), with some subfields gaining and others losing in standing. The status of subfields is related to different publishing habits, but some of those differences are opposite to those observed when considering the hierarchy of scientific fields. We examine possible explanations for the ``status'' of different subfields. Some natural explanations  availability of funding, importance of applications  do not appear to function, suggesting that factors internal to the discipline are at work. We propose a different type of explanation, based on a notion of ``focus'' of a subfield, that might or might not be specific to mathematics. 
Properness for circle packings and Delaunay circle patterns on complex projective structures.
JeanMarc Schlenker and Andrew Yarmola.
arXiv:1806.05254..
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. 
Delaunay Triangulations of Points on Circles.
Vincent Despré, Olivier Devillers, Hugo Parlier, JeanMarc Schlenker
arXiv:1803.11436.
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a maxmin angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasilinear time by a very simple algorithm. 
Notes on the Schwarzian tensor and measured foliations at infinity of
quasifuchsian manifolds.
JeanMarc Schlenker.
arXiv:1708.01852.
The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. \par We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincar\'eEinstein metrics, the other (specifically for conformally flat structures) of the second fundamental form of a hypersurface in a "constant curvature" space with a degenerate metric, interpreted as the space of horospheres in hyperbolic space. This clarifies a relation between linear Weingarten surfaces in hyperbolic manifolds and MongeAmp\`ere equations.
Notes aiming at clarifying the relations between different points of view and introducing one new notion, no real result. Not intended to be submitted at this point 
Higher signature Delaunay decompositions.
Jeffrey Danciger, Sara Maloni, JeanMarc Schlenker.
arXiv:1602.03865.
Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semiRiemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the wellknown optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a wellknown description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.
Some questions on antide Sitter geometry. Thierry Barbot, Francesco Bonsante, Jeff Danciger, William M. Goldman, François Guéritaud, Fanny Kassel, Kirill Krasnov, JeanMarc Schlenker, Abdelghani Zeghib. arXiv:1205.6103.
We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature $1$. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichmüller theory, as well as analogs in hyperbolic geometry.