Variation and rigidity of quasi-local mass

Abstract

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.

arXiv commentnew notations added; references updated; section 4 revised

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Cited by in corpus (2)

• On the evolution of the spacetime Bartnik mass
• Quasi-local Penrose inequalities with electric charge