Convex real projective structures and Hilbert metrics

Abstract

Abstract. 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 metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in SL(3, R). In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.

Holomorphic differentials, thermostats and Anosov flows

“…An oriented properly convex projective surface is an example of a surface carrying a (G, X)-structure where X = S 2 is the oriented projective 2-sphere and G = SL(3, R) its group of projective transformations, cf. [21]. In particular, it follows that the path geometry associated to (g, A) is flat.…”

“…Consequently, we obtain a (two-dimensional) divisible convex set. Since Ω is convex, it is equipped with the Hilbert metric and moreover, the Hilbert metric descends to define a Finsler metric on the quotient surface M ≃ Ω/Γ, see in particular [21] for a nice survey of these ideas. We observe that the geodesic flow of the Finsler metric is a C 1 reparametrisation of the flow we associate to the pair (g, A).…”

Extremal conformal structures on projective surfaces

“…In particular, (5.1) is known as Wang's equations in the affine sphere literature [43]. We refer the reader to the survey articles [23], [33] as well as [1] for additional details.…”

Optimal Inequalities Between Distances in Convex Projective Domains

“…As a special case, we obtain the Blaschke metric (or Cheng-Yau metric) if the immersion is a proper affine hypersphere with mean curvature −1 which is asymptotic to the boundary ∂K Ω . The Blaschke metric gives rise to the Blaschke distance d B on Ω (see [9], [2], [7], [8]).…”