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# Advances in Theoretical and Mathematical Physics

## Volume 23 (2019)

### Number 5

### Variation and rigidity of quasi-local mass

Pages: 1411 – 1426

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n5.a5

#### Authors

#### Abstract

Inspired by the work of Chen–Zhang [5], we derive an evolution formula for theWang-Yau quasi-local energy in reference to a static space, introduced by Chen–Wang–Wang–Yau [4]. 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 [9], 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.

The second-named author’s research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105.

Published 12 February 2020