TY - JOUR

T1 - Variation and rigidity of quasi-local mass

AU - Lu, Siyuan

AU - Miao, Pengzi

N1 - Funding Information:
The second named author’s research was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #281105.

PY - 2019

Y1 - 2019

N2 - 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 ∑, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasilocal mass at ∑. 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.

AB - 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 ∑, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasilocal mass at ∑. 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.

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U2 - 10.4310/ATMP.2019.v23.n5.a5

DO - 10.4310/ATMP.2019.v23.n5.a5

M3 - Article

AN - SCOPUS:85083774489

VL - 23

SP - 1411

EP - 1426

JO - Advances in Theoretical and Mathematical Physics

JF - Advances in Theoretical and Mathematical Physics

SN - 1095-0761

IS - 5

ER -