Holomorphic cubic differentials and convex domains in the real projective plane

The Cheng-Yau affine sphere in R^3 projectivizing to a given proper convex domain in RP^2 endows the domain with a canonical holomorphic cubic differential. While Labourie and Loftin deduced from this construcution a holomorphic parametrization of the SL(3,R)-Hitchin component of a closed surface, I will explain how a flat infinite sector of the cubic differential gives rise to a line segment on the boundary of the convex domain, which implies a holomorphic parametrizations of the space of certain convex projective structures on an open surface, generalising the result of Dumas and Wolf. I will also discuss problems arising from the analogy between this construction and harmonic maps to the hyperbolic plane.