Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 5

Variation and rigidity of quasi-local mass

Pages: 1411 – 1426



Siyuan Lu (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada)

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)


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