A Liouville theorem for the real twisted Monge-Ampère equation
Abstract
We establish a Pogorelov type interior C2 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(D2u) 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.
Related
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- Quadratic growth rigidity for the complex Monge--Ampère equation
- Lp Brunn-Minkowski inequality for the eigenvalue of generalized Monge-Ampère equation
BibTeX
@misc{jordan2026liouville,
title = {{A Liouville theorem for the real twisted Monge-Ampère equation}},
author = {Joshua Jordan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00543},
eprint = {2610.00543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00543},
}
RIS
TY - UNPB
TI - A Liouville theorem for the real twisted Monge-Ampère equation
AU - Joshua Jordan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00543
UR - https://arxiv.org/abs/2610.00543
AB - 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.
ER -