Convex real projective structures on closed surfaces are closed

Abstract:

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