Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $\Sigma$, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasi-local mass at $\Sigma$. Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in \cite{Lu-Miao}, we prove a rigidity theorem for compact $3$-manifolds with nonnegative scalar curvature, with boundary. This rigidity theorem in turn gives a characterization of the equality case of the localized Penrose inequality in $3$-dimension.

中文翻译：

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准局部质量的变化和刚度

受 Chen-Zhang \cite{Chen-Zhang} 工作的启发，我们推导出了参考静态空间的 Wang-Yau 准局域能量的演化公式，由 Chen-Wang-Wang-Yau \cite{CWWY 引入}. 如果参考静态空间表示初始表面 $\Sigma$ 的质量最小化、静态扩展，我们观察到 Wang-Yau 准局域能量的导数等于 Bartnik 准局域质量在 $\西格玛$。将准局部能量的演化公式与\cite{Lu-Miao} 中证明的局部彭罗斯不等式相结合，我们证明了具有非负标量曲率、有边界的紧致$3$-流形的刚性定理。该刚性定理反过来给出了 $3$ 维中局部彭罗斯不等式的等式情况的表征。