Determine the available size of the maximal development for barrier construction

Determine how large the vacuum maximal development of an asymptotically flat initial Cauchy surface is in the regions relevant to constructing a maximal hypersurface, so that the barrier conditions required by the usual elliptic construction of maximal foliations can be verified.

Background

The paper explains that constructing a maximal hypersurface by standard elliptic methods requires suitable barrier conditions in relevant regions of the ambient vacuum spacetime. However, the size of the vacuum maximal development is not known a priori, making it difficult to determine whether those barriers exist in the necessary regions. This uncertainty is identified as a fundamental obstacle to applying the usual elliptic approach when the initial Cauchy surface is not maximal. The paper instead develops a mean-curvature-flow strategy to construct maximal foliations under smallness assumptions on the initial mean curvature.