Research in differential geometry at IMAG, in the GTA team.

I belong to the French research groups RT2182 Tétraèdre (previously GDR Platon) and to the ANR project GAPR.

Teaching at the mathematics department of Faculté des Sciences de Montpellier.

News

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Check Antoin's first article Affine Deformations of Divisible Convex Cones and Affine Spacetimes

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Cours CIMPA/Écoles Mathématiques Africaines Divers aspects des surfaces Université d’El-Oued, Algérie, November 02-14 2025

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Earlier

2023-...

Publications

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Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces,
with Roman Prosanov
Journal für die reine und angewandte Mathematik 2025 article

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An Introduction to Reshetnyak’s Theory of Subharmonic Distances,
in Reshetnyak's Theory of Subharmonic Metrics, CRM-Springer 2023 article arxiv 

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Hyperbolic geometry of shapes of convex bodies,
with Clément Debin.
Groups, Geometry, and Dynamics 2022. article

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A remark on one-harmonic maps from a Hadamard surface of pinched negative curvature to the hyperbolic plane,
with Andrea Seppi.
Josai Mathematical Monographs 2021 article

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Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces,
with Andrea Seppi.
Annales Henri Lebesgue 2020 article

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Quasi-Fuchsian co-Minkowski manifolds,
with Thierry Barbot.
in In the tradition of Thurston, Springer Cham, 2020 article arxiv

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A note on invariant constant curvature immersions in Minkowski space,
with Graham Smith,
Geometriae Dedicata, 2020 article arxiv

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Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds,
with Dmitriy Slutskiy,
Mathematische Zeitschrift, 2019 article arxiv

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Spherical, hyperbolic and other projective geometries: convexity, duality, transitions,
with Andrea Seppi,
in Eighteen Essays in Non-Euclidean Geometry, 2019 article arxiv

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Group actions and scattering problems in Teichmüller theory,
with Graham Smith,
in Handbook of group action IV, 2018 article arxiv

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The equivariant Minkowski problem in Minkowski space,
with Francesco  Bonsante.
Ann. Inst. Fourier , 2017 article

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A remark on spaces of flat metrics with cone singularities of constant sign curvatures,
with Ivan Izmestiev.
Sémin. Théor. Spectr. Géom. 2017 article

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A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies
Sémin. Théor. Spectr. Géom. 2017 article

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Shapes of polyhedra, mixed volumes, and hyperbolic geometry,
with Ivan Izmestiev.
Mathematika, 2017 article arxiv

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Lorentzian area measures and the Christoffel problem,
with Giona Veronelli.
Ann. Sc. Norm. Super. Pisa, 2016 article

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Hyperbolization of cusps with convex boundary,
with Ivan Izmestiev and Giona Veronelli.
Manuscripta Math., 2016 article arxiv

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Christoffel and Minkowski problems in Minkowski space,
Théor. Spectr. Géom. 2015 article

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Polygons of the Lorentzian plane and spherical simplexes,
Elem. Math., 2014 article

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Fuchsian convex bodies: basics of Brunn--Minkowski theory,
Geom. Funct. Anal. 2013 article arxiv

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Flippable tilings of constant curvature surfaces,
with Jean-Marc Schlenker.
Illinois J. Math., 2012 article

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Gauss images of hyperbolic cusps with convex polyhedral boundary, with Ivan Izmestiev.
Trans. Am. Math. Soc., 2011 article

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Fuchsian polyhedra in Lorentzian space-forms,
Mathematische Annalen, 2011 article arxiv

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From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston,
Enseign. Math., 2011 article

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Existence and uniqueness theorems for convex polyhedral metrics on compact surfaces,
in Current problems in mathematics and mechanics. Vol. VI, 2010 arxiv

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Hyperbolic cusps with convex polyhedral boundary,
with Ivan Izmestiev.
Geometry and Topology, 2009 article

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Polyhedral hyperbolic metrics on surfaces,
Geom. Dedicata, 2008 Erratum Geom. Dedicata 2009 article arxiv

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Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces,
Ann. Inst. Fourier, 2007 article

Supervisor

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Antoine Ablondi, PhD Montpellier / Turin (with Andrea Seppi) 2024-
Affine Deformations of Divisible Convex Cones and Affine Spacetimes arxiv June 2024

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Hicham Labeni, PhD Cergy 2016-2020
Realizing metrics of curvature \leq −1 on closed surfaces in Fuchsian anti-de Sitter manifolds
Journal of the Australian Mathematical Society, 2021 article arxiv

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Dmitriy Slutskiy, Post doc Cergy 2015-2016
Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds
Mathematische Zeitschrift, 2019 article arxiv

Vitae

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I graduated in 2006 under the supervisation of Bruno Colbois (Neuchâtel) and Jean-Marc Schlenker (Toulouse ---now Luxembourg). I spent one year post doc position in Friboug (CH) under the supervision of Ruth Kellerhals I was a Maître de Conférence at Université de Cergy-Pontoise (now called CYU), then I moved to Université de Montpellier in 2020 on the basis of an agreement, and then formally in 2022.

I spent the academic year 2019-2020 at the IRL CRM-CNRS at Université de Montréal

During the Covid lockdown, we ran the online Pangolin Seminar (2020-2021) with some colleagues.

Corganizer of the weekly seminar Séminaire Darboux at Montpellier, 2022-2025.