AI 中文总结
针对k≥n/2的σ_k曲率方程，通过建立σ_k的新凹性不等式，结合谱变量分解方法，得到闭星形k-凸超曲面及欧氏Hessian方程的全局曲率估计。
AI 中文摘要
我们为初等对称函数σ_k建立了一个新的凹性不等式，该不等式控制由logλ_max的二阶变分产生的谱二次型。证明基于谱变量的易-难分解：易区域使用σ_k的最优约束凹性估计处理，难区域通过Gårding根坐标及逆根有序部分和的凹性分析。作为应用，当n/2≤k<n时，我们得到满足σ_k(κ)=f(X,ν)（f为一般正右端项）的闭星形k-凸超曲面的全局曲率估计，以及欧氏Hessian方程对应的全局到边界估计。
英文摘要
We establish a new concavity inequality for the elementary symmetric function \(σ_k\), which controls the spectral quadratic form arising from the second variation of \(\logλ_{\max}\). The proof is based on an easy--hard decomposition of the spectral variables. The easy region is treated using an optimal constrained concavity estimate for \(σ_k\), whereas the hard region is analyzed through Gårding-root coordinates and the concavity of the ordered partial sums of the inverse roots. As an application, for \(n/2\leq k<n\), we obtain global curvature estimates for closed star-shaped \(k\)-convex hypersurfaces satisfying \(σ_k(κ)=f(X,ν)\) with a general positive right-hand side, together with the corresponding global-to-boundary estimates for Euclidean Hessian equations.
Comments40 pages
