Preprint
Liouville Rigidity for Real and Complex Degenerate Hessian Equations
arXiv (Cornell University)
arXiv
07/23/2026
DOI: 10.48550/arxiv.2607.21024
Abstract
We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set𝓐 . The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globallyC^(0,α)entire viscosity solution of \[ Hess_(F)uın \] is constant if and only if𝓐is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate Gårding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standardk -Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.
Details
- Title: Subtitle
- Liouville Rigidity for Real and Complex Degenerate Hessian Equations
- Creators
- Hao FangBiao MaJinyang Wu
- Resource Type
- Preprint
- Publication Details
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2607.21024
- ISSN
- 2331-8422
- Publisher
- arXiv
- Language
- English
- Date posted
- 07/23/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985183640802771
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