Lower semicontinuity of mass under C0{C^{0}} convergence and Huisken’s isoperimetric mass

Abstract

Abstract Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>C</m:mi><m:mn>0</m:mn></m:msup></m:math> {C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.