Study of hyperbolic directions in convex projective geometry.
problem Understanding properties of quasi-geodesics in convex projective geometry.
method Three perspectives: Hilbert metric, boundary projective geometry, and automorphisms.
result Relationship between different definitions of Morse and regular quasi-geodesics.
The paper characterizes groups acting on real projective spaces.
problem Understanding groups acting on convex domains in real projective geometry.
method Proves structure theorem for relatively hyperbolic groups in real projective spaces.
result Characterizes groups in terms of invariant convex subsets.
Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.
problem Understanding Hilbert geometry over general valued fields and their limits.
method Developed a theory of Hilbert geometry over general ordered valued fields, proving ultralimit results.
result Ultralimit of rescaled real Hilbert geometries is isometric to a non-Archimedean Hilbert metric space.
The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.
problem Characterizing and understanding geometric properties of 4D projective manifolds.
method Analyzing geometric decompositions and using properties of locally symmetric spaces.
result Closed, indecomposable 4D projective manifolds are either real hyperbolic or have real hyperbolic pieces.
The paper defines equations for constructing projective structures on 3-manifolds.
problem Constructing projective structures on 3-manifolds.
method Defining a system of real valued equations and inequalities.
result These equations can be used to detect properly convex structures on 3-manifolds.
We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good pr…
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…
We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metr…
The paper defines conditions for groups acting on convex domains to be relatively hyperbolic.
problem Understanding conditions for groups acting on convex domains to be relatively hyperbolic.
method Analyzing the geometry of the convex domain to determine relative hyperbolicity.
result Established necessary and sufficient conditions for groups to be relatively hyperbolic.
We construct a compact convex generating set Cn of the moduli set of closed connected projective special real manifolds of fixed dimension n. We show that a closed connected projective special real manifold corresponds to an inner point of Cn if and only if it has regular boundary behaviour.…
The study connects polygon areas and projective structures in 3D space.
problem Relating polygon areas and projective structures in 3D space.
method Investigates positive tuples of complete flags in R^3 and their associated polygons in RP^2.
result Establishes a relationship between Holmes-Thompson area and projective structures.
Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial result…
There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests…
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
Proves a higher rank rigidity theorem for convex real projective manifolds.
problem No specific problem stated; focuses on proving a theorem.
method Analogue of Ballmann and Burns-Spatzier's higher rank rigidity theorem.
result Proves a higher rank rigidity theorem for convex real projective manifolds.
The Funk metric connects billiards, projective geometry, and convex geometry.
problem Exploring the Funk metric's invariants and inequalities.
method Using the Funk metric, extending results from projective geometry and convex geometry.
result General affine inequalities and volume maximizers in Funk geometry.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the real-projective structure as the conformal structure is fixed and the cubic diffe…
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Symplectic coordinates found on projective structures on orbifolds.
problem Symplectic structure on deformation spaces of convex projective structures.
method Global Darboux coordinates system construction and symplectic space decomposition.
result Symplectic form on deformation space of convex projective structures.
Affine deformations of convex cones on projective surfaces.
problem Understanding affine actions on convex cones.
method Geometric correspondence and convex tube domains.
result Quotients of convex domains are affine manifolds with convex surfaces.
A real projective orbifold is an n-dimensional orbifold modeled on RPn with the group PGL(n+1,R). We concentrate on an orbifold that contains a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed (n−1)-dimensional orbifolds times …
Study real projective structures on a specific Coxeter orbifold.
problem Characterize real projective structures on a noncompact Coxeter orbifold.
method Embedding and extending a Coxeter quadrilateral, perturbing to form a convex polytope, and analyzing the deformation space.
result Determine the detailed properties of the deformation space of real projective structures on the orbifold.
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …
A theorem of Tits - Vinberg allows to build an action of a Coxeter group Γ on a properly convex open set Ω of the real projective space, thanks to the data P of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…
Study on the limits of projective special real manifolds and their symmetries.
problem Understanding the limits of projective special real manifolds.
method Evolution of defining polynomial and centro-affine fundamental form along curves.
result Found a list of possible limit geometries and a lower bound for symmetry groups.
In this paper we study the degeneration of convex real projective structures on bordered surfaces.
We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
Proof that convex structures on manifolds are open and closed.
problem Characterizing the set of holonomies for convex structures.
method Expository proof using properties of holonomies and projective structures.
result The set of holonomies forms an open and closed subset of the space of representations.
The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
The paper disproves some implications in convex projective geometry.
problem Geometrical finiteness in round convex projective geometry.
method Construction of counterexamples and description of invariant domains.
result Existence of a counterexample with infinite Hilbert volume.
Study on projective orbifolds with ends and their deformation theory.
problem Understanding deformation theory of projective orbifolds with ends.
method Analysis of convex real projective structures on strongly tame n-orbifolds.
result Deformation theory for nicest projective orbifolds.
Properly convex manifolds with generalized cusps have irreducible holonomy.
problem Characterizing the holonomy of properly convex real-projective manifolds.
method Analyzing ends of manifolds and their fundamental groups.
result Holonomy of properly convex manifolds is irreducible under certain conditions.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture…
New geometric transitions studied via real algebra degenerations.
problem Understanding geometric transitions not arising from limits of ambient geometries.
method New degenerations of complex hyperbolic space and construction of new geometries over real algebras.
result Generalization of geometric transitions to new constructions over real algebras.
The paper explores volume product and slicing conjectures using convex body deformations.
problem Volume product and slicing conjectures in convex geometry.
method Study of variational aspects of volume product functional under projective deformations.
result Provides a proof of a theorem by Klartag and identifies critical convex bodies.