The semicontinuity phenomenon of the ADM mass under pointed (i.e.,
local) convergence of asymptotically flat metrics is of interest because
of its connections to nonnegative scalar curvature, the positive mass
theorem, and Bartnik’s mass-minimization problem in general relativity.
We extend a previously known semicontinuity result in dimension three for
pointed convergence to higher dimensions, up through seven, using recent work of S.
McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality
of H. Bray and D. Lee). For a technical reason, we restrict to the case in
which the limit space is asymptotically Schwarzschild. In a separate result,
we show that semicontinuity holds under weighted, rather than pointed,
convergence, in
all dimensions
,
with a simpler proof independent of the positive mass theorem. Finally, we also
address the two-dimensional case for pointed convergence, in which the asymptotic
cone angle assumes the role of the ADM mass.
