Affine complete locally convex hypersurfaces

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The Monge–Ampère equation and its link to optimal transportation (2013, background): “Another classical instance where the Monge-Ampère equation arises is affine geometry, more precisely in the “affine sphere problem” and the “affine maximal surfaces” problem (see for instance [28, 98, 30, 107, 108, 109]).”
General affine surface areas (2009, background): “Numerous important results regarding equi-affine surface area were obtained in recent years (see, for example, [1,2,45,48–51]).”
Survey on Affine Spheres (2008, background): “In fact, affine completeness implies Euclidean completeness for hypersurfaces in R for n ≥ 2 by [71], and so we have the following result, also proved by Jia-Li [43]:”
Polynomial cubic differentials and convex polygons in the projective plane (2014, background): “Let h = eu|dz|2 solve the Wang equation (6) and suppose that eu|dz|2 is a complete metric on C.”
The Bernstein problem for affine maximal hypersurfaces (2000, background): “Much efforts were done toward this conjecture (see for examples [2, 4, 8, 24, 41] for partial results), until a cornerstone paper by Trudinger-Wang [40].”