Global convex solutions to det D²u = periodic measure decompose uniquely (up to constant) into quadratic polynomial plus periodic function.
Figalli,The Monge-Amp `ere equation and its applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨urich
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A Liouville theorem for convex functions with periodic Monge-Amp\`ere measure
Global convex solutions to det D²u = periodic measure decompose uniquely (up to constant) into quadratic polynomial plus periodic function.