end $i_c$.[]{data-label="fig:kerr-extrem-diag"}](kerr-extremo-color-p.pdf){width="5cm"}

![On the left, an the initial data with two asymptotically flat ends, like the non-extreme Kerr black holes. For these data the strict inequality holds. On the right, the data of extreme Kerr black hole, with one asymptotically flat and one cylindrical end. For this data the equality holds.[]{data-label="fig:non-extreme-id"}](kerr-initial-data-iso-p.pdf "fig:"){width="30.00000%"} ![On the left, an the initial data with two asymptotically flat ends, like the non-extreme Kerr black holes. For these data the strict inequality holds. On the right, the data of extreme Kerr black hole, with one asymptotically flat and one cylindrical end. For this data the equality holds.[]{data-label="fig:non-extreme-id"}](kerr-ex-initial-data-iso-p.pdf "fig:"){width="35.00000%"}

Regarding the quasi-local inequality (\[eq:JAd\]) we have the following result.

\[t:2\] Given an axisymmetric closed marginally trapped and stable surface ${\Sigma}$, in a spacetime with non-negative cosmological constant and fulfilling the dominant energy condition, it holds the inequality $$\label{eq:JAt}
8\pi |J| \leq  A \quad (= \text{Extreme Kerr throat}),$$ where $A$ and $J$ are the area and angular momentum of ${\Sigma}$.

This is a pure spacetime and local result. That is, there is no mention of a three-dimensional initial hypersurface where the two-dimension surface ${\Sigma}$ is embedded. Axisymmetry is only imposed on ${\Sigma}$. Moreover, this theorem does not assume 