Abstract
Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed
A Equivalence of definitions of isoperimetric mass
Recall Definition 11, in which
Proposition 37.
Let
where the supremum is taken over all sequences
In other words, defining the isoperimetric mass using exhaustions is equivalent to using sequences whose perimeters become arbitrarily large.
Proof.
We only need to prove that
Note that
where “big O” depends on W but not on i. Using the isoperimetric inequality to bound the volume
Inequality (A.1) now follows. From this inequality, we conclude that
Acknowledgements
The authors thank Gerhard Huisken and Felix Schulze for helpful discussions and the referee for thoughtful comments.
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