# Variation and rigidity of quasi-local mass

## TL;DR

This paper derives an evolution formula for Wang-Yau quasi-local energy, linking it to Bartnik mass, and uses it to prove a rigidity theorem characterizing equality cases in a localized Penrose inequality for 3-manifolds.

## Contribution

It introduces a new evolution formula for quasi-local energy and establishes a rigidity theorem connecting quasi-local mass concepts with geometric inequalities.

## Key findings

- Derivative of Wang-Yau energy equals Bartnik mass derivative under certain conditions
- Rigidity theorem for 3-manifolds with nonnegative scalar curvature
- Characterization of equality cases in localized Penrose inequality

## 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.