Surveys in Differential Geometry

Volume 19 (2014)

Rigidity and minimizing properties of quasi-local mass

Pages: 49 – 61



Po-Ning Chen (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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


In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects that the mass should vanish. We describe the Wang-Yau quasi-local mass whose definition is motivated by this rigidity property and by the Hamilton-Jacobi analysis of the Einstein-Hilbert action. In addition, we survey recent results on the minimizing property the Wang-Yau quasi-local mass.


general relativity, quasi-local energy

2010 Mathematics Subject Classification

53C50, 83C40

Published 6 March 2015