Advances in Theoretical and Mathematical Physics
Volume 23 (2019)
Variation and rigidity of quasi-local mass
Pages: 1411 – 1426
Inspired by the work of Chen–Zhang , we derive an evolution formula for theWang-Yau quasi-local energy in reference to a static space, introduced by Chen–Wang–Wang–Yau . 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 , 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