Convex real projective structures on closed surfaces are closed

Abstract

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 ) .