AI 中文总结
本文利用共形散度恒等式，在正爱因斯坦背景下建立了$\sigma_2$-曲率的平均曲率估计、刚性、尖锐不等式与稳定性，推广了多个经典结果。
AI 中文摘要
利用带有可变参考曲率的经典共形散度恒等式，我们证明了四个主要结果。首先，我们在$A_g\in\overline{\Gamma_2^+}$且具有正的规定$H_2$数据的圆半球上，为共形度量建立了一个一般的平均曲率估计。当$\sigma_2(A_g)=0$且边界数据非递增时，该估计得出刚性，去除了Case-Wang的夹逼条件$\sup_\Sigma H_g\le 3\inf_\Sigma H_g$。对于$n\ge 5$，常数数据情形也对与他们的尖锐$\sigma_2$ Sobolev迹猜想相关的光滑临界度量进行了分类。其次，在正爱因斯坦共形类中，我们将Viaclovsky和Gursky-Streets的常数$\sigma_2$刚性结果推广到非递增规定数据，包括$n\ge 5$时具有非零Weyl曲率的背景。第三，我们将Li-Li的球面$\sigma_2/\sigma_1$刚性和Guan-Wang的尖锐积分不等式推广到正爱因斯坦背景，后者在正数量曲率下对$n\ge 5$成立。第四，我们将Frank-Peteranderl的球面$\sigma_2$稳定性推广到$n\ge 5$维中固定的非圆正爱因斯坦背景，在正数量曲率下保留$H^1$和$W^{1,4}$控制。
英文摘要
Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{Γ_2^+}$ and positive prescribed $H_2$ data. When $σ_2(A_g)=0$ and the boundary data are nonincreasing, the estimate yields rigidity, removing Case-Wang's pinching condition $\sup_ΣH_g\le 3\inf_ΣH_g$. For $n\ge 5$, the constant-data case also classifies the smooth critical metrics associated with their sharp $σ_2$ Sobolev trace conjecture. Second, within positive Einstein conformal classes, we extend the constant-$σ_2$ rigidity results of Viaclovsky and Gursky-Streets to nonincreasing prescribed data, including backgrounds with nonzero Weyl curvature for $n\ge 5$. Third, we extend Li-Li's spherical $σ_2/σ_1$ rigidity and Guan-Wang's sharp integral inequality to positive Einstein backgrounds, with the latter holding for $n\ge 5$ under positive scalar curvature. Fourth, we extend Frank-Peteranderl's spherical $σ_2$ stability to fixed nonround positive Einstein backgrounds in dimensions $n\ge 5$, retaining $H^1$ and $W^{1,4}$ control under positive scalar curvature.
