报告题目：Liouville Rigidity for Real and Complex Degenerate Hessian Equations

报告人：吴金洋，上海科技大学

报告时间：7月1日，09:00-10:00

报告地点：第五教学课5501

摘要：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 $\mathcal{A}$. The main structural notion is \emph{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 $C^{0,\alpha}$ entire viscosity solution of $\mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A}$ is constant \emph{if and only if} $\mathcal{A}$ is Liouville admissible; thus the Liouville property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate G{\aa}rding 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. This talk is based on joint work with Hao Fang and Biao Ma.