Existence of entire solutions to the Monge-Ampère equation

We prove the existence of infinitely many entire convex solutions to the Monge-Amp\`ere equation ${\rm det} D^2 u=f$ in ${\Bbb R}^n$, assuming that the inhomogeneous term $f\ge 0$ and is of polynomial growth at infinity. When $f$ satisfies the doubling condition, we show that solution is of polynomial growth. As an application, we resolve the existence of translating solutions to a class of Gauss curvature flow.

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