发表机构
University of Iowa; East China Normal University; ShanghaiTech University（爱荷华大学; 华东师范大学; 上海科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入行列式椭圆性条件，连接了紧致Kähler流形上完全非线性椭圆算子的$L^\infty$估计与代数组合性质，并通过行列式容量正性刻画了Gårding椭圆算子的估计准则。
AI 中文摘要
我们引入行列式椭圆性（det-ellipticity），这是紧致Kähler流形上完全非线性椭圆算子的一种定量结构条件，它为$L^\infty$-估计与一大类Hessian椭圆算子的代数/组合性质之间提供了联系。在解析方面，我们引入$\mathcal D$-子解，推广了Sui--Sun的行列式子水平条件。利用Guo--Phong--Tong的辅助比较方法和复Monge--Ampère方程的位势理论，我们在粘度上解和奇异参考位势的行列式-熵界下，获得了大上同调类中的相对$L^\infty$-估计。Det-椭圆性为这类子解提供了系统构造。在代数方面，我们研究度为$d$的Gårding椭圆多项式算子。我们通过最高齐次部分的行列式容量的正性来刻画指数为$d/n$的行列式增量界。这等价于秩一标量化的Newton多面体的平衡点条件，以及相关子空间多拟阵的斜率半稳定性。结合解析假设，这些准则为相应的完全非线性方程提供了相对$L^\infty$估计。
英文摘要
We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce $\mathcal D$-subsolutions extending the determinant sublevel condition of Sui--Sun. Using the auxiliary comparison method of Guo--Phong--Tong and pluripotential theory for complex Monge--Ampère equations, we obtain relative $L^\infty$-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree $d$. We characterize the determinant increment bound with exponent $d/n$ by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative $L^\infty$ estimates for the corresponding fully nonlinear equations.
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