Families of non time-symmetric initial data sets and Penrose-like energy inequalities

Motivated by solving the constraint equations in the evolutionary form suggested by Rácz, we propose a family of asymptotically flat initial data sets which are “asymptotically spherically symmetric” at infinity. Within this family, we obtain Penrose-like energy estimates and establish the existence of solutions for the constraint equations in the spherical symmetric and totally umbilic cases.