arXiv research

Study how convex structures on surfaces change.

The paper surveys partial results on convex real projective orbifolds with specific ends.

Study real projective structures on a specific Coxeter orbifold.

Proof that convex structures on manifolds are open and closed.

The paper characterizes groups acting on real projective spaces.

Survey on deformations of convex structures on manifolds and orbifolds.

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. 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 on projective orbifolds with ends and their deformation theory.

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 flows on convex real projective structures on surfaces.

The moduli space of projective structures is tessellated using Euclidean cell decompositions.

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…

The paper defines equations for constructing projective structures on 3-manifolds.

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

We prove that the space of convex real projective structures on a surface of genus $g\ge 2$ admits a mapping class group invariant Kähler metric where Teichmüller space with Weil-Petersson metric is a totally geodesic complex submanifold.

New examples of hyperbolic manifolds with convex projective structures.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.

We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.

Study on entropy of bulging deformations of projective structures on surfaces.

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…

The study proves a local rigidity theorem for hyperbolic lattices using real projective structures.

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…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

Study convex projective manifolds with compact and convex ends.

A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to …

The study connects polygon areas and projective structures in 3D space.

Proves a higher rank rigidity theorem for convex real projective manifolds.

Notes on Fock and Goncharov's moduli spaces of real projective structures.

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…

Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.

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…

The paper studies the correlation of Hilbert lengths for convex projective surfaces.

Let M be a compact surface of negative Euler characteristic and let C(M) be the deformation space of convex real projective structures on M. For every choice of pants decomposition for M, there is a well known parameterization of C(M) known as the Goldman parameterization. In this paper, we study how some geometric pro…

We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with r…

Study convex cocompact subgroups in real projective geometry.

Researchers describe coordinates for a space of convex RP² structures.

We try to understand the geometric properties of $n$-manifolds ($n\geq 2$) with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e., $n$-manifolds with projectively flat torsion free affine connections. We define the notion of $i$-convexity of such manifolds due to Carriére for integers $i$, $1 \leq i \le…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …

Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.

The paper shows how certain projective representations act on convex domains.

3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.

This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Arch…

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.

Properly convex manifolds with generalized cusps have irreducible holonomy.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

Compact set identifies projective special real manifolds.

The paper proposes a framework for structured prediction using projection oracles.