Communications in Analysis and Geometry

Volume 31 (2023)

Number 7

On the moduli space of asymptotically flat manifolds with boundary and the constraint equations

Pages: 1849 – 1866

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n7.a8

Authors

Sven Hirsch (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Martin Lesourd (Black Hole Initiative, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

Carlotto–Li have generalized Marques’ path connectedness result for positive scalar curvature $R \gt 0$ metrics on closed $3$-manifolds to the case of compact $3$-manifolds with $R \gt 0$ and mean convex boundary $H \gt 0$. Using their result, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $\mathbb{R}^3 \setminus B^3$ is path connected. The argument bypasses Cerf’s theorem, which was used in Marques’ proof but which becomes inapplicable in the presence of a boundary. We also show path connectedness for a class of maximal initial data sets with marginally outer trapped boundary.

Full Text (PDF format)

Received 2 March 2020

Accepted 2 September 2021

Published 10 August 2024