Lorentzian area and volume estimates for integral mean curvature bounds

In the present paper we establish area and volume estimates for spacetimes satisfying the strong energy condition in terms of the area and the $L^n$-norm of the second fundamental form or the mean curvature of an initial Cauchy hypersurface. We believe that these estimates will lay some of the groundwork in establishing new convergence results for Cauchy developments $(M_j, g_j)$ of suitably converging initial data $(\Sigma_j ,h_j ,K_j )$.