arXiv subjects

Antoine Ablondi

Publications and source records attributed to Antoine Ablondi.

3 recordsLinked to original sources

arXiv2026

A De Rham Perspective on the Symplectic Geometry of Teichmüller Space

We present a new proof of Wolpert's Magic Formula, stating that any Fenchel--Nielsen coordinates on the Teichmüller space of a closed surface are Darboux coordinates for the Weil--Petersson symplectic form. Our approach relies on a de Rham cohomology model for the tangent space to the character variety model of Teichmüller space, in which, by the seminal work of Goldman, the Weil--Petersson form is expressed by a natural symplectic form, called the Goldman form. We extend the work of Fillastre and Seppi, who have managed, using that approach and Stokes's Theorem, to provide a new proof of Wolpert's sum of cosines formula. Using the unique isometric symmetry on any hyperbolic pair of pants, we also introduce a way, given a pair of pants decomposition of a closed surface, to construct an associated linear involution on every tangent space of its Teichüller space. That allows us to derive a new proof of Wolpert's Magic Formula and deduce a self-contained proof of the closedness of the Goldman form on Teichmüller space.

math.GT↗

arXiv2026

Affine Deformations of Divisible Convex Cones and Affine Spacetimes

Let $G$ be a subgroup of $\mathrm{SL}(\mathbb{R}^{d+1})\ltimes\mathbb{R}^{d+1}$ obtained by adding a translation part to a torsion-free discrete subgroup of $\mathrm{SL}(\mathbb{R}^{d+1})$ dividing a convex cone in the sense of Benoist. We consider the maximal convex domains in $\mathbb{R}^{d+1}$ on which the affine action of $G$ is free and properly discontinuous, and show its quotient by $G$ is naturally endowed with an "affine spacetime" structure, which is a generalisation of the notion of flat Lorentzian spacetime. More precisely, we show that this quotient is a Maximal Globally Hyperbolic affine spacetimes admitting a $C^2$ locally uniformly Convex and Compact Cauchy surface (denoted as a MGHCC affine spacetimes), and that it comes with a cosmological time function with Cauchy hypersurfaces foliating the quotient affine spacetime as level sets. Finally, we show such quotients are the only examples of MGHCC affine spacetimes. All these results generalise the work of Mess, Barbot and Bonsante on affine deformations of uniform lattices of $\mathrm{SO}_0(d,1)$.

math.DG↗

arXiv2026

An Affine Invariant Minkowski Problem

In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $μ$ on the unit sphere $\mathbb{S}^d$, to find a compact convex set $K$ with area measure $μ$. For convex sets in the Minkowski space invariant under an affine deformation of a uniform lattice of $\mathrm{SO}_0(d,1)$, the analogous Minkowski problem was considered and solved by Barbot--Béguin--Zeghib (partially) and Bonsante--Fillastre. By a theorem of Mess--Barbot--Bonsante, that also solves the Minkowski problem in flat Lorentzian spacetimes with compact hyperbolic Cauchy surface. We consider convex domains of the oriented real affine space $\mathbb{R}^{d+1}$ which are invariant under a subgroup of affine transformations obtained by adding translation parts to a discrete subgroup of $\mathrm{SL} (\mathbb{R}^{d+1})$ dividing a convex cone. We prove that those convex domains satisfy a local Steiner Formula, allowing to introduce natural area measures and define an affine invariant Minkowski problem. We then solve that Minkowski problem through a variational method using the convexity of a covolume functional. We also give an interpretation of those results in some "affine spacetimes", which were introduced by the author in a preceding work.

math.DG↗