A Liouville theorem for the real twisted Monge-Ampère equation

Joshua Jordan

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arXiv · 2610.00543

A Liouville theorem for the real twisted Monge-Ampère equation

Joshua Jordan

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Abstract

We establish a Pogorelov type interior $C^2$ estimate for the real twisted Monge-Ampere equation without assuming uniform ellipticity. This equation is fully nonlinear and elliptic, but is neither concave nor a function of the eigenvalues of the Hessian, so the standard techniques for interior second-derivative estimates do not directly apply. Building on the partial Legendre transform formulation of Streets and Warren, we derive a differential identity for the associated positive-definite matrix $W(D^2u)$ and use some linear algebra to control the third-order terms arising in the maximum principle argument. As an application, we prove a Liouville theorem for entire split convex solutions with quadratic growth, extending the rigidity theorem of Streets and Warren by replacing their uniform ellipticity hypothesis with suitable quadratic growth assumptions.

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Joshua Jordan. 2026-09-30. A Liouville theorem for the real twisted Monge-Ampère equation. https://arxiv.org/abs/2610.00543

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