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

Weisong Dong

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

Analysis of PDEs 2026-02-24 v1

Authors: Weisong Dong

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.

Keywords

Liouville theorem Monge-Ampere equation concavity inequality interior estimates quadratic growth

Cite

@article{arxiv.2602.20090,
  title  = {A Liouville-type theorem for $2$-Monge-Amp\`ere equation in dimension three},
  author = {Weisong Dong},
  journal= {arXiv preprint arXiv:2602.20090},
  year   = {2026}
}

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18pages

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