Given a bounded convex domain $\Omega$ in $RP^n$, Cheng-Yau provide a unique solution to the Monge-Ampere equation $\det v_{ij} = (-1/v)^{n+2}$, for convex $v$ and Dirichlet boundary value $v=0$. This solution $v$ then determines a unique hypersurface, a hyperbolic affine sphere, asymptotic to the boundary of the cone over $\Omega$ in $R^{n+1}$. The hyperbolic affine sphere carries tensors, the Blaschke metric and cubic form, which are invariant under special linear automorphisms of the cone. These tensors descend to $\Omega$ to be projectively invariant. Benoist-Hulin
show that if $\Omega_i\to\Omega$ in the Hausdorff topology, then these tensors converge in the local $C^\infty$ topology.

A 2-dimensional hyperbolic affine sphere carries a conformal structure induced by the Blaschke metric, and the cubic tensor is equivalent to a
holomorphic cubic differential. A quotient of such a domain $\Omega$ by a subgroup of $PGL(3,R)$ acting discretely and properly discontinuously then is called a real projective surface, and the conformal structure and cubic differential descend to the quotient. We will discuss a recent result
relating degenerations of convex real projective surfaces along necks in terms of the geometry of the bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces. In addition to the Benoist-Hulin convergence results mentioned above, the proof also uses analytic techniques of Dumas-Wolf and Wolpert.