Jean-Marc Schlenker

Recent preprints

On (2,1)-dimensional anti-de Sitter spacetimes with convex boundary. Roman Prosanov, Jean-Marc Schlenker, Graham Andrew Smith. arXiv:2610.05506
Let S be a closed oriented surface of genus at least 2. Let X be a (2,1)-dimensional anti-de Sitter spacetime homeomorphic to S × [−1, 1] with smooth, spacelike, strongly convex boundary. By Gauss’ formula, the induced metric on each of the two components of ∂X has curvature K < −1. In this paper we address the inverse problem. We show that, for any two smooth metrics h₋, h₊ on S with curvature K < −1, there exists a unique, smooth anti-de Sitter metric on X := S × [−1, 1] with respect to which the boundary is locally strongly convex and has induced metric isotopic to h₋ on its past component and isotopic to h₊ on its future component. Existence has already been proven by Tamburelli.
We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations l₋, l₊ ∈ ℳℒ_S that together fill S, there exists a unique (2,1)-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively l₋ and l₊. Here, existence has already been proven by Bonsante–Schlenker.

Equivariant isometric immersions of surfaces in hyperbolic space. Jean-Marc Schlenker. arXiv:2609.32300
Let S be a closed, oriented surface of genus at least 2, let h be a smooth Riemannian metric on S with curvature K ∈ (−1, 0], and let c ∈ 𝒯_S be a conformal structure on S. There exists a unique equivariant isometric immersion of (S, h) in ℍ³ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is c. Dually, if h^∗ is a smooth metric on S with curvature K^∗ ∈ (−∞, 0) and if c ∈ 𝒯_S, there exists a unique equivariant immersion of S into ℍ³ with third fundamental form h^∗ and such that the pull-back of the conformal class at infinity by the Gauss map is c.
Equivalently, given h and c ∈ 𝒯_S, there is a unique pair (E, u) where E is a hyperbolic end with conformal structure at infinity c and u is an isometric embedding of (S, h) in E. Given h^∗ and c, there exists a unique pair (E, u^∗), where E is a hyperbolic end with conformal structure at infinity c, and u^∗ : S → E is an embedding inducing the third fundamental form h^∗. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

Projective rigidity of circle patterns and polyhedral surfaces in hyperbolic ends. Jean-Marc Schlenker. arXiv:2508.15339
Let S be a closed, orientable surface of genus g ≥ 2. We consider Delaunay circle patterns on S equipped with a complex projective structure. We prove that the space of complex projective structures on S equipped with a Delaunay circle pattern of prescribed combinatorics and intersection angles is a manifold of dimension 6g − 6, and that the forgetful map to the space 𝒞_S of ℂP¹-structures on S is a Lagrangian immersion. This extends a recent result of Bonsante and Wolf for circle packings. This statement, and its proof, are more conveniently stated in terms of ideal polyhedral surfaces (surfaces with vertices at infinity) in hyperbolic ends, with the angles between the circles corresponding to the dihedral angles. Seen from this angle, we extend the statement to ideal polyhedral surfaces with prescribed edge lengths (or induced metrics), and to other types of polyhedral surfaces, either compact or hyperideal.

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 S × ℝ, 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.

The Weyl problem for unbounded convex domains in ℍ³. Jean-Marc Schlenker. arXiv:2106.02101
Let K ⊂ ℍ³ be a convex subset in ℍ³ with smooth, strictly convex boundary. The induced metric on ∂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 ℍ³. The induced metric on ∂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 ℍ³, together with the induced metric on ∂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 ℍ³. 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.

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.

Unpublished manuscripts

Those manuscripts are not intended for publication.

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 equipped 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).
We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poincaré–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–Ampère 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

Some questions on anti-de Sitter geometry. Thierry Barbot, Francesco Bonsante, Jeff Danciger, William M. Goldman, François Guéritaud, Fanny Kassel, Kirill Krasnov, Jean-Marc 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.