In this paper, we study complete convex solutions of certain nonlinear elliptic equations by using geometric methods. We present a proof of the Jörgens–Calabi–Pogorelov theorem about improper convex affine spheres based on the study of complete convex solutions of the simplest Monge–Ampère equation. We consider a similar problem for Monge–Ampère equations of more general types. We prove that, under certain assumptions, solutions of these equations are quadratic polynomials.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 147, Proceedings of the Workshop on Algebra and Geometry of the Samara University, 2018.
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Kokarev, V.N. Complete Convex Solutions of Monge–Ampère-Type Equations and their Analogs. J Math Sci 248, 303–337 (2020). https://doi.org/10.1007/s10958-020-04874-2
Keywords and phrases
- improper convex affine sphere
- Monge-Ampère equation
AMS Subject Classification