Communications in Analysis and Geometry

Volume 30 (2022)

Number 4

Quasi-local mass on unit spheres at spatial infinity

Pages: 745 – 778



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.)

Ye-Kai Wang (Department of Mathematics, National Cheng Kung University, Tainan City, Taiwan)

Shing-Tung Yau (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


In this note, we compute the limit of the Wang–Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang–Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a vacuum spacetime, the quasi-local mass decays faster and the leading order term is related to the Bel–Robinson tensor. Several new techniques of evaluating quasilocal mass are developed in this note.

P.-N. Chen is supported by NSF grant DMS-1308164, and by Simons Foundation collaboration grant #584785.

M.-T. Wang is supported by NSF grants DMS-1405152 and DMS-1810856.

Y.-K. Wang is supported by MOST Taiwan grants 105-2115-M-006-016-MY2 and 107-2115-M-006-001-MY2.

S.-T. Yau is supported by NSF grants PHY-0714648 and DMS-1308244.

The authors would like to thank the National Center for Theoretical Sciences at National Taiwan University where part of this research was carried out.

Received 23 March 2019

Accepted 9 September 2019

Published 30 January 2023