A Liouville-type theorem for $2$-Monge-Ampère equation in dimension three

TL;DR

This paper proves a Liouville-type theorem for smooth, entire solutions $u:\mathbb{R}^3\to\mathbb{R}$ of the 2-Monge–Ampère equation $m_2(\lambda(D^2u))=1$ under a cone condition $\lambda(D^2u)\in P_2^{1/2}$ and a quadratic-growth assumption. The authors first establish a crucial concavity (Jacobi-type) inequality for the operator in $\mathbb{R}^3$, enabling a Pogorelov-type interior $C^2$ estimate. With this interior control and an Evans–Krylov argument, they deduce global regularity that forces the Hessian to be constant, hence $u$ must be a quadratic polynomial. This work extends classical Liouville results for Monge–Ampère and Hessian quotient equations to the $2$-Monge–Ampère setting in dimension three, under a natural cone constraint. The approach combines precise structural identities for $m_2$, a tailored concavity inequality, and a robust interior estimate to achieve rigidity for entire solutions.

Abstract

We prove that every entire solution with quadratic growth, lying in a suitable cone, to the 2-Monge-Ampère 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.

Theorems & Definitions (10)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• proof