Affine spheres and convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-manifolds

We outline consequences of a theorem of Cheng-Yau in affine differential geometry for manifolds locally modeled on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. In particular, for properly convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-manifolds, we observe that there are two canonical projectively flat connections and a canonical Riemannian metric. One connection represents the given [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-structure and the other the [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]-structure of the projective dual manifold. When n is 2, we use this approach together with a result of C.-P. Wang to show that a compact oriented convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /]-surface of genus at least two is equivalent to a conformal structure on the surface together with a holomorphic tricanonical form. This recovers a theorem of Goldman on the deformation space of such surfaces, and yields a description of the moduli space.