Liouville Rigidity for Real and Complex Degenerate Hessian Equations

2026/07/23 by Hao Fang, Biao Ma, Jinyang Wu
#math.AP #math.DG

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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 set A. 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, globally C^0,α entire viscosity solution of Hess_{\mathbb F}u∈\partialA is constant if and only if A is 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 standard k-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.

Cited by

• Ideal Gårding polynomials

Related

• Liouville theorem for a class of Hessian equations
• Liouville property and existence of entire solutions of Hessian equations
• Viscosity solution to complex Hessian quotient equations
• Viscosity solution to complex Hessian equations on compact Hermitian manifolds
• Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations