Real Projective Structures Research Articles

Ideal triangulations and geometric transitions

Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti-de Sitter (AdS) geometry. Our approach involves solving Thurston's gluing equations over several different shape parameter algebras. In the case of a punctured torus bundle with Anosov monodromy, we identify two components of real solutions for which there are always nearby positively oriented solutions over both the complex and pseudo-complex numbers. These complex and pseudo-complex solutions define hyperbolic and AdS structures that, after coordinate change in the projective model, may be arranged into one continuous family of real projective structures. We also study the rigidity properties of certain AdS structures with tachyon singularities.

Real projective structures on a real curve

Given a compact connected Riemann surface X equipped with an antiholomorphic involution τ, we consider the projective structures on X satisfying a compatibility condition with respect to τ. For a projective structure P on X, there are holomorphic connections and holomorphic differential operators on X that are constructed using P. When the projective structure P is compatible with τ, the relationships between τ and the holomorphic connections, or the differential operators, associated to P are investigated. The moduli space of projective structures on a compact oriented C∞ surface of genus g≥2 has a natural holomorphic symplectic structure. It is known that this holomorphic symplectic manifold is isomorphic to the holomorphic symplectic manifold defined by the total space of the holomorphic cotangent bundle of the Teichmüller space Tg equipped with the Liouville symplectic form. We show that there is an isomorphism between these two holomorphic symplectic manifolds that is compatible with τ.

Projective deformations of hyperbolic Coxeter 3-orbifolds

By using Klein’s model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev’s theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.

Erratum to: Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds

Erratum to: Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds

Actions of discrete groups on spheres and real projective spaces

In this paper, we first define discrete, smooth actions on $$ \mathbb{S}^{{2n + 1}} $$ , whose limit sets are Cantor sets wildly embedded in $$ \mathbb{S}^{{2n + 1}} $$ (Antoine's necklaces). Secondly, we define Schottky groups on real projective spaces of odd dimensions, $$ \mathbb{P}^{{2n + 1}}_{\mathbb{R}} $$ . We lift these actions to (locally) projective actions on the sphere $$ \mathbb{S}^{{2n + 1}} $$ and consider the quotient space of the domain of discontinuity by the group to obtain new examples of manifolds with real projective structures.

Convex projective structures on Gromov–Thurston manifolds

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.

Flat metrics, cubic differentials and limits of projective holonomies

F. Labourie and the author independently have shown that a convex real projective structure on an oriented closed surface S of genus at least two is equivalent to a pair of a conformal structure plus a holomorphic cubic differential. Along certain paths, we find the limiting holonomy of convex real projective structures on a surface S corresponding corresponding to a given fixed conformal structure S and a holomorphic cubic differential λ U 0 as \(\lambda\to\infty\) . We explicitly give part of the data needed to identify the boundary point in Inkang Kim’s compactification of the deformation space of convex real projective structures. The proof follows similar analysis to that studied by Mike Wolf is his application of harmonic map theory to reproduce Thurston’s boundary of Teichmüller space.

Flat Projective Structures on Surfaces and Cubic Holomorphic Differentials

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a minimal surface in a symmetric space.

The deformation spaces of convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-structures on 2-orbifolds

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 notes on hyperbolic 2-orbifolds, the previous work of Goldman on convex real projective structures on surfaces, and some classical geometry.

Compactification of Strictly Convex Real Projective Structures

In this paper we compactify the space of convex real projective structures, known as Hitchin’s component in a representation variety. We give geometric meanings to the boundary points.

The Compactification of the Moduli Space of Convex ℝℙ2 Surfaces, I

There is a canonical identification, due independently to the author and to F. Labourie, 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 a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of Goldman. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of Cheng–Yau and C.P. Wang and a result of Dunkel on the asymptotics of systems of ODEs.

Solvariétés projectives de dimension trois

Let V be a hyperbolic 2-torus bundle over the circle, or equivalently, a quotient of the three-dimensional group Sol by a lattice Γ. We study real projective structures on V. We prove that all such structures are left-invariant which means that they are all obtained by the following process: take a representation of Sol in PGL (4, R) with an open orbit Ω in RP3; it induces a projective structure on Sol and, hence, on the left-quotient Γ\Sol = V. We classify all the real projective structures on V and show that they are in fact all affine. The proof involves a study of codimension one projective subsurfaces, which are tori, and the dynamic of their holonomy group in the whole manifold.

Rigidity and Deformation Spaces of Strictly Convex Real Projective Structures on Compact Manifolds

In this paper we show that if two strictly convex, compact real projective manifolds have the same marked length spectrum with respect to the Hilbert metric, then they are projectively equivalent. This is a rigidity for Finsler metric with a special geometric structure. Furthermore we prove an analogue of a Hitchin's conjecture for hyperbolic 3-manifolds, namely the deformation space of convex real projective structures on a compact hyperbolic 3-manifold M is a component in the moduli space of PGL(4,ℍ)-representations of π1(M).

The convex and concave decomposition of manifolds with real projective structures

The convex and concave decomposition of manifolds with real projective structures

The classification of real projective structures on compact surfaces

Choi gratefully acknowledges partial support from GARC-KOSEF. Goldman gratefully acknowledges partial support from the National Science Foundation, the Alfred P. Sloan Foundation and the Institute for Advanced Computer Studies at the University of Maryland.

I-Convexity of Manifolds with Real Projective Structures

i-Convexity of Manifolds with Real Projective Structures

𝑖-convexity of manifolds with real projective structures

We compare the notion of higher-dimensional convexity, as defined by Carrière, for real projective manifolds with the existence of hemispheres. We show that if an i-convex real projective manifold M of dimension n for an integer i with 0 > i > n 0 > i > n has an i-dimensional hemisphere, then M is projectively homeomorphic to S n / Γ {{\mathbf {S}}^n}/\Gamma where Γ \Gamma is a finite subgroup of O ( n + 1 , R ) O(n + 1,{\mathbf {R}}) acting freely on S n {{\mathbf {S}}^n} .

Convex Real Projective Structures on Closed Surfaces are Closed

Convex Real Projective Structures on Closed Surfaces are Closed

Convex real projective structures on closed surfaces are closed

The deformation space C ( Σ ) \mathfrak {C}(\Sigma ) of convex R P 2 \mathbb {R}{{\mathbf {P}}^2} -structures on a closed surface Σ \Sigma with χ ( Σ ) > 0 \chi (\Sigma ) > 0 is closed in the space Hom ⁡ ( π , SL ⁡ ( 3 , R ) ) / SL ⁡ ( 3 , R ) \operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R}) of equivalence classes of representations π 1 ( Σ ) → SL ⁡ ( 3 , R ) {\pi _1}(\Sigma ) \to \operatorname {SL} (3,\mathbb {R}) . Using this fact, we prove Hitchin’s conjecture that the contractible "Teichmüller component" (Lie groups and Teichmüller space, preprint) of Hom ⁡ ( π , SL ⁡ ( 3 , R ) ) / SL ⁡ ( 3 , R ) \operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R}) precisely equals C ( Σ ) \mathfrak {C}(\Sigma ) .

Convex real projective structures on compact surfaces

Convex real projective structures on compact surfaces