Variation and rigidity of quasi-local mass

Abstract

Inspired by the work of Chen-Zhang [5], we derive an evolution formula for the Wang-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 Σ, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasi-local mass at Σ.Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in [10], 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.

Rigidity of Riemannian Penrose inequality with corners and its implications

“…In the case of Riemannian Penrose inequality with corners, the rigidity part was studied by Shi, Wang and Yu [21] in 3-dimension, and the manifolds were shown to be static with zero scalar curvature and, under an additional geometric condition, (Ω, g Ω ) was proven to be isometric to a region in (M m , g m ). Theorem 1.1 was also proved by the authors [13] for the case n = 3, in the setting of the localized Penrose inequality [12].…”

A quasi-local Penrose inequality for the quasi-local energy with static references

“…In [14], it is also proved that the equality of the quasi-local Penrose inequality implies that the two mean curvatures are equal. See also [7,15,22] on the equality case of this quasi-local Penrose inequality.…”

On the evolution of the spacetime Bartnik mass

“…In [11], the Wang-Yau mass with reference to static spaces was introduced by P.-N Chen, M.-T. Wang, Y.-K. Wang, and S.-T. Yau. In the time-symmetric setting, recent work by S. Lu and the second-named author [14] indicates that, if the static metric extension conjecture holds, the derivative of the Bartnik mass along an evolving family of surfaces agrees with the derivative of the Wang-Yau mass with reference to the static metric extension of the given surface. In making this observation, the derivative formula of the Bartnik mass in time-symmetric initial data (see [17]) plays a key role.…”