Convex real projective structures on closed surfaces are closed

Abstract

Abstract. The deformation space C(L) of convex RP2-structures on a closed surface I with #(5)) < 0 is closed in the space Hom(7t, SL(3, R))/SL(3, R) of equivalence classes of representations nx (Z) -» SL(3, R). Using this fact, we prove Hitchin's conjecture that the contractible "Teichmiiller component" (Lie groups and Teichmiiller space, preprint) of Homfw, SL(3, R))/SL(3, R) precisely equals C(2).Let X be a closed orientable surface of genus g > 1 and n = nx(L) its fundamental group. A convex EP2-structure on M is a representation (uniformization) of M as a quotient £l/T where Q c RP2 is a convex domain and T c SL(3, R) is a discrete group of collineations of RP2 acting properly and freely on Q. (See [5] for basic theory of such structures.) The space of projective equivalence classes of convex RP2-structures embeds as an open subset in the space of equivalence classes of representations n -* SL(3, R). The purpose of this note is to show that this subset is also closed.In [7], Hitchin shows that the space of equivalence classes of representations n -> SL(3, R) falls into three connected components: one component C_i consisting of classes of representations for which the associated flat R3-bundle over £ has nonzero second Stiefel-Whitney class; a component Co containing the class of the trivial representation; a component Cx diffeomorphic to a cell of dimension 16(g -1), which he calls the "Teichmiiller component." While C-i can be distinguished from Co and Cx by a topological invariant [3, 4], no characteristic invariant distinguishes representations in the Teichmiiller component from those in Co. The Teichmiiller component is defined as follows. Using the Klein-Beltrami model of hyperbolic geometry, a hyperbolic structure on Z is a special case of a convex RP2-structure Q/T where Q is the region bounded by a conic. In this case I" is conjugate to a cocompact lattice in SO(2,l) c SL(3, R). The space T(S) of hyperbolic structures ("Teichmiiller

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

“…This component, known as the Teichmüller or Hitchin component, has special geometric significance. In the case of SL(3, R), Choi and Goldman [9,10] showed that the representations in the Hitchin component are discrete and faithful and correspond to convex projective structures on the surface. More recently, Labourie [26] has shown for G = SL(n, R) that these representations are discrete and faithful and are related to certain Anosov geometric structures, and analogous results have been obtained for G = Sp(2n, R) by Burger et al [6].…”

Automorphismes des cônes convexes

“…Remarque En particulier, tout sous-groupe Zariski dense de SL(2, R) contient unélément à valeurs propres négatives (ce fait est démontré dans [10]) et il existe des sous-groupes Zariski dense de SL(3, R) à valeurs propres positives (par exemple, les groupes d'holonomie des structures projectives convexes sur les surfaces construits dans [13]). …”

Minimal Lagrangian connections on compact surfaces

“…Properly convex projective surfaces are of particular interest as the may be seen -through the work of Hitchin [8] and Choi & Goldman [4] -as the natural generalisation of the notion of a hyperbolic Riemann surface. Labourie [9] and Loftin [11] have independently shown that on a compact surface of negative Euler characteristic there exists a one-to-one correspondence between properly convex projective structures and pairs (J, C) consisting of a complex structure J and a cubic differential C that is holomorphic with respect to J.…”