Jean-Marc Schlenker

Recent preprints

Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus. Roman Prosanov, Jean-Marc Schlenker. arXiv:2502.01599
We prove that globally hyperbolic compact anti-de Sitter (2+1)-spacetimes with strictly convex spacelike boundary that is either smooth or polyhedral and whose holonomy is close to Fuchsian are determined by the induced metric on the boundary.

Weakly almost-Fuchsian manifolds are nearly-Fuchsian. Manh-Tien Nguyen, Jean-Marc Schlenker, Andrea Seppi. arXiv:2501.12277
We show that a hyperbolic three-manifold M containing a closed minimal surface with principal curvatures in [−1,1] also contains nearby (non-minimal) surfaces with principal curvatures in (−1,1). When M is complete and homeomorphic to SxR, for S a closed surface, this implies that M is quasi-Fuchsian, answering a question left open from Uhlenbeck's 1983 seminal paper. Additionally, our result implies that there exist (many) quasi-Fuchsian manifolds that contain a closed surface with principal curvatures in (−1,1), but no closed minimal surface with principal curvatures in (−1,1), disproving a conjecture from the 2000s.

Filling Riemann surfaces by hyperbolic Schottky manifolds of negative volume. Tommaso Cremaschi, Viola Giovannini, Jean-Marc Schlenker. arXiv:2405.07598
We provide conditions under which a Riemann surface X is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on X enough closed curves of short enough hyperbolic length.

Convex co-compact hyperbolic manifolds are determined by their pleating lamination. Bruno Dular, Jean-Marc Schlenker. arXiv:2403.10090
Convex co-compact 3-dimensional hyperbolic manifolds are uniquely determined by the pleating measured lamination on the boundary of their convex core.

The Weyl problem for unbounded convex domains in H³. Jean-Marc 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.

The prestige and status of research fields within mathematics. Jean-Marc 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 over-represented in the most selective mathematics journals or among recipients of top prizes. Moreover this status of subfields evolves markedly over the period of observation (1984--2016), 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. Jean-Marc 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, Jean-Marc 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 max-min 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 quasi-linear time by a very simple algorithm.

Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds. Jean-Marc 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\'e-Einstein 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 Monge-Amp\`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, Jean-Marc 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 semi-Riemannian 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 well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.

Some questions on anti-de Sitter geometry.

Thierry Barbot

Francesco Bonsante

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.