Communications in Analysis and Geometry

Volume 28 (2020)

Number 7

Quasi-local energy with respect to de Sitter/anti-de Sitter reference

Pages: 1489 – 1531



Po-Ning Chen (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)

Mu-Tao Wang (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [7, 26, 27] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti‑de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.

P.-N. Chen is supported by NSF grant DMS-1308164, M.-T. Wang is supported by NSF grants DMS-1105483 and DMS-1405152, and S.-T. Yau is supported by NSF grants PHY-0714648 and DMS-1308244. This work was partially supported by a grant from the Simons Foundation (#305519 to Mu-Tao Wang).

Received 11 March 2016

Accepted 5 March 2018

Published 7 December 2020