An Extension to a Theorem of Jörgens, Calabi, and Pogorelov

(1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analytical proof, along the lines of affine geometry, of the theorem was later given by Cheng and Yau [9]. The first author extended the result for classical solutions to viscosity solutions [4]. It was proven by Trudinger and Wang in [19] that the only open convex subset of Rn which admits a convex C2 solution of det(D2u) = 1 in with limx→∂ u(x) = ∞ is = Rn . In this paper we give the following extension to the theorem of Jörgens, Calabi, and Pogorelov: Let u be a convex viscosity solution of det(D2u) = 1 outside a bounded subset of