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François Fillastre

Publications and source records attributed to François Fillastre.

At least 19 recordsLinked to original sources
arXiv2026

An explicit exotic representation of a rank-one simple Lie group via convex bodies

In [DP12], Delzant and Py showed that there exist continuous irreducible isometric actions of $\mathrm{PSL}_2(\mathbb{R})$ on the infinite-dimensional hyperbolic space $\mathbb{H}^\infty$. Such continuous irreducible actions do not exist on the hyperbolic spaces $\mathbb{H}^n$ when $n>2$ and their associated embeddings $\mathbb{H}^2 \to \mathbb{H}^\infty$ given by the orbit maps were later called \emph{exotic} by Monod and Py in [MP14]. In this article, we produce a continuous and irreducible representation of $\mathrm{PSL}_2(\mathbb{R})\to \mathrm{Isom}(\mathbb{H}^\infty)$ using the hyperbolic model for convex bodies introduced in [DF22]. This yields a convex cocompact $\mathrm{PSL}_2(\mathbb{R})$-action on the infinite-dimensional hyperbolic space $\mathbb{H}^\infty$, of which the compact quotient over the minimal $\mathrm{PSL}_2(\mathbb{R})$-invariant convex set is homeomorphic to the 2-dimensional oriented Banach--Mazur compactum. Moreover, we study the geometry of one of its orbit maps and compute the Hausdorff dimension of the limit set of this representation.

math.GR↗
arXiv2025

Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any flat metric on $S$ with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally hyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is given by $h$. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the hyperbolic surface, from which the flat metric can be easily recovered. We show next that given two such flat metrics on the surface, there exists a unique pair of future- and past-complete flat globally hyperbolic maximal (2+1)-spacetimes with the same holonomy, in which the flat metrics embed respectively as convex polyhedral Cauchy surfaces. The proof follows from convexity properties of the total length of the associated balanced geodesic cellulations over Teichmüller space.

math.MG↗
arXiv2022

An introduction to Reshetnyak's theory of subharmonic distances

The aim of the present text is to provide some basics around Reshetnyak's theory of subharmonic distances, together with an overview of the main results. This text is intended to be a complement to the English translation of \cite{R1954}, \cite{Res1959}, \cite{R60I}, \cite{R60II}, \cite{R61}, \cite{R61b}, \cite{R62}, \cite{R63}, \cite{R63III} and \cite{huber}. They all will be published in a same volume \cite{livre}. While subharmonic distances are often confused with distances of bounded curvature, we present them as a complete autonomous theory. In turn, there is no specific prerequisite for the present text.

math.DG↗
arXiv2021

Hyperbolic geometry of shapes of convex bodies

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov--Fenchel Inequality is interpreted as the Lorentzian reversed Cauchy--Schwarz Inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by $-1$ (in the sense of Alexandrov). In dimension $3$, this space is homeomorphic to the space of distances with non-negative curvature on the $2$-sphere, and this latter space contains the space of flat metrics on the $2$-sphere considered by W.P.~Thurston. Both Thurston's and the area distances rely on the area form. So the latter may be considered as a generalization of the "real part" of Thurston's construction.

math.DG↗
arXiv2020

Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.

math.GT↗
arXiv2019

A note on invariant constant curvature immersions in Minkowski space

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a unique affine deformation $Γ:=(ρ,τ)$ of a Fuchsian group such that $(S,g_+)$ and $(S, g_-)$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $Γ$ of an open subset $Ω$ of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Min\-kow\-ski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which $(S, g_+)$ and $(S, g_-)$ are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.

math.DG↗
arXiv2019

Spherical, hyperbolic and other projective geometries: convexity, duality, transitions

Since the end of the 19th century, and after the works of F. Klein and H. Poincaré, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a "limit" of both geometries. Then all the geometries that can be obtained in this way. Some of these geometries had a rich development, most remarkably hyperbolic geometry and the Lorentzian geometries of Minkowski, de Sitter and anti-de Sitter spaces, which in higher dimension have had large interest for a long time in mathematical physics and more precisely in General Relativity. Moreover, some degenerate spaces appear naturally in the picture, namely the co-Euclidean space (the space of hyperplanes of the Euclidean space), and the co-Minkowski space (that we will restrict to the space of space-like hyperplanes of Minkowski space), first because of duality reasons, and second because they appear as limits of degeneration of classical spaces. In fact, co-Minkowski space recently regained interest under the name half-pipe geometry. The purpose of the present paper is to provide a survey on the properties of these spaces, especially in dimensions 2 and 3, from the point of view of projective geometry. Even with this perspective, the paper does not aim to be an exhaustive treatment. Instead it is focused on the aspects which concern convex subsets and their duality, degeneration of geometries and some properties of surfaces in three-dimensional spaces. The presentation is intended to be elementary, hence containing no deep proofs of theorems, but trying to proceed by accessible observations and elementary proofs.

math.DG↗
arXiv2018

Quasi-Fuchsian co-Minkowski manifolds

This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there is a convex core. There is also a unique "mean" hypersurface, i.e. with traceless second fundamental form. The mean distance between the mean hypersurface and the lower boundary of the convex core endows the space of affine deformations of a given lattice with an asymmetric norm. The symmetrization of the asymmetric norm is simply the volume of the convex core. In dimension 2+1, the asymmetric norm is the total length of the bending lamination of the lower boundary component of the convex core. We obtain an extrinsic proof of a theorem of Thurston saying that, on the tangent space of Teichmüller space, the total length of measured geodesic laminations is an asymmetric norm. We also exhibit and comment the Anosov-like character of these deformations, similar to the Anosov character of the quasi-Fuchsians representations pointed out in Guichard--Wienhard.

math.DG↗
arXiv2017

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a local structure of complex pseudo-spheres. The complex structure comes again from the area of the metric. The form can be degenerate; its signature depends on the curvatures prescribed. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.

math.DG↗
arXiv2017

A short elementary proof of reversed Brunn--Minkowski inequality for coconvex bodies

The theory of coconvex bodies was formalized by A.~Khovanski{\uı} and V.~Timorin in \cite{KT}. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint \cite{schnei2}, R.~Schneider proved a result that implies a reversed Brunn--Minkowski inequality for coconvex bodies, with description of equality case. In this note we show that this latter result is an immediate consequence of a more general result, namely that the volume of coconvex bodies is strictly convex. This result itself follows from a classical elementary result about the concavity of the volume of convex bodies inscribed in the same cylinder.

math.MG↗
arXiv2017

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $ϕ$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $ϕ(η)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $η$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $μ$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $μ$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $Γ$ of isometries of $\mathbb{H}^d$, given a $Γ$ invariant Radon measure $μ$, given a isometry group $Γ_τ$ of Minkowski space, with $Γ$ as linear part, there exists a unique convex set with area measure $μ$, invariant under the action of $Γ_τ$. The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Ampère equation. The existence part can be translated as an existence result for Monge--Ampère equation. The regular version was proved by T.~Barbot, F.~Béguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $Γ_τ=Γ$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $Γ_τ$-invariant surface of constant Gauss-Kronecker curvature equal to $1$.

math.DG↗
arXiv2016

Group actions and scattering problems in Teichmüller theory

In recent years, Teichmüller theory, which is the study of moduli spaces of marked Riemann surfaces, has come to be considered more and more from the point of view of actions of surface groups inside certain semi-simple Lie groups. In particular, we consider the case where the Lie groups in question have symmetric spaces which are lorentzian spacetimes. Indeed, this can be considered as the starting point of Mess' seminal work, which led to the development of new and strikingly simpler proofs of many results of Teichmüller theory by considering them in terms of geometric objects inside these symmetric spaces. Our aim is to provide a brief and straightforward introduction to this approach, whilst developing what we consider to be a useful mental framework for organising known results and open problems.

math.DG↗
arXiv2015

Hyperbolization of cusps with convex boundary

We prove that for every metric on the torus with curvature bounded from below by -1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.

math.MG↗
arXiv2015

Lorentzian area measures and the Christoffel problem

We introduce a particular class of unbounded closed convex sets of $\R^{d+1}$, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature $(+,...,+,-)$ instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space $\H^d$. Important examples are embeddings of the universal cover of so-called globally hyperbolic maximal flat Lorentzian manifolds. Basic tools are first derived, similarly to the classical study of convex bodies. For example, F-convex sets are determined by their support function, which is defined on $\H^d$. Then the area measures of order $i$, $0\leq i\leq d$ are defined. As in the convex bodies case, they are the coefficients of the polynomial in $ε$ which is the volume of an $ε$ approximation of the convex set. Here the area measures are defined with respect to the Lorentzian structure. Then we focus on the area measure of order one. Finding necessary and sufficient conditions for a measure (here on $\H^d$) to be the first area measure of a F-convex set is the Christoffel Problem. We derive many results about this problem. If we restrict to "Fuchsian" F-convex set (those who are invariant under linear isometries acting cocompactly on $\H^d$), then the problem is totally solved, analogously to the case of convex bodies. In this case the measure can be given on a compact hyperbolic manifold. Particular attention is given on the smooth and polyhedral cases. In those cases, the Christoffel problem is equivalent to prescribing the mean radius of curvature and the edge lengths respectively.

math.DG↗
arXiv2013

Polygons of the Lorentzian plane and spherical simplexes

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.

math.DG↗
arXiv2010

Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when $S$ is the sphere are classical.

math.DG↗