A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres

Abstract

We use constructions by Miao and Chruściel-Delay to produce asymptotically flat metrics on R 3 \mathbb {R}^3 which have zero scalar curvature and multiple stable minimal spheres. Such metrics are solutions of the time-symmetric vacuum constraint equations of general relativity, and in this context the horizons of black holes are stable minimal spheres. We also note that under pointwise sectional curvature bounds, asymptotically flat metrics of nonnegative scalar curvature and small mass do not admit minimal spheres, and hence are topologically R 3 \mathbb {R}^3 .