momentum is zero and hence theorem \[t:1\] reduces to the positive mass theorem. In order to have non-zero angular momentum we need to allow non-trivial topologies, for example manifolds with two asymptotic ends as it is the case in theorem \[t:1\]. An important initial data set that satisfies the hypothesis of the theorem is provided by an slice $t=constant$ in the Kerr black hole in the standard Boyer-Lindquist coordinates, see figures \[fig:kerr-diag\] and \[fig:kerr-extrem-diag\]. The non-extreme initial data have a different geometry as the extreme initial data. The former are asymptotically flat at both ends. In contrast, extreme initial data, which reach the equality in (\[eq:60\]), have one asymptotically flat end and one cylindrical end, see figure \[fig:non-extreme-id\]. That geometry represents the “optimal shape” with respect to the inequality (\[eq:60\]). Figure \[fig:non-extreme-id\] is the analog of figure \[fig:1\] for the geometrical inequality (\[eq:60\]).

![Conformal diagram of the non-extreme Kerr black hole. The points $i_0$ represent spacelike infinity. The surface $S$ have two identical asymptotically flat ends $i_0$.[]{data-label="fig:kerr-diag"}](kerr-color-p.pdf){width="6cm"}

![Conformal diagram of the extreme Kerr black hole. The point $i_0$ represents spacelike infinity, the point $i_c$ represent the cylindrical end. The surface $S$ has one asymptotically flat end $i_0$ and one cylindrical 