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, NC 27708-0320, USA)

Martin Lesourd (Black Hole Initiative, Harvard University, Cambridge, MA 02138, USA)

Abstract

Carlotto-Li have generalized Marques' path connectedness result for positive scalar curvature $R>0$ metrics on closed $3$-manifolds to the case of compact $3$-manifolds with $R>0$ and mean convex boundary $H>0$. Using their result, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $\mathbb{R}^{3}\backslash B^{3}$ 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 26 July 2024