A Liouville-type theorem for $2$-Monge-Amp\`ere equation in dimension three

TL;DR

This paper proves that entire solutions with quadratic growth to the 2-Monge-Ampère equation in three dimensions are quadratic polynomials, using concavity inequalities and interior estimates.

Contribution

It establishes a Liouville-type theorem for the 2-Monge-Ampère equation in three dimensions, extending understanding of solution classification.

Findings

Solutions with quadratic growth are quadratic polynomials.

The proof uses concavity inequalities and Pogorelov-type estimates.

The result applies to solutions lying in a suitable cone.

Abstract

We prove that every entire solution with quadratic growth, lying in a suitable cone, to the 2-Monge-Amp\`ere equation on $\mathbb{R}^3$ is a quadratic polynomial. The proof proceeds by first establishing a concavity inequality, and then deriving a Pogorelov-type interior $C^2$ estimate.