this kind of equality (or a related inequality) to the dynamical regime.

Theorems {#sec:theorems}
========

The Penrose inequality $$\label{eq:9}
  \sqrt{\frac{A}{16\pi}} \leq m \quad (=\text{Schwarzschild}),$$ has been intensively studied. It is a very relevant geometric inequality for black holes since it is valid without any symmetry assumption. For a comprehensive review on this subject see [@Mars:2009cj]. The most important results concerning this inequality are the proofs of Huisken-Ilmanen [@Huisken01] and Bray [@Bray01] for the Riemannian case. The general case remains open. Also, there is up to now no result concerning the Penrose inequality with angular momentum (\[eq:pendj\]) discussed in the previous section.

In the following we present a sample of the main results concerning inequalities (\[eq:mjdd\]) and (\[eq:JAd\]) that have been recently proved.

For the global inequality (\[eq:mjdd\]) we have the following theorem.

\[t:1\] Consider an axially symmetric, vacuum, asymptotically flat and maximal initial data set with two asymptotics ends. Let $m$ and $J$ denote the total mass and angular momentum at one of the ends. Then, the following inequality holds $$\label{eq:60}
\sqrt{|J|} \leq  m \quad (= \text{Extreme Kerr}).$$

For the precise definitions, fall off conditions an assumptions on the initial data we refer to original articles cited bellow.

The first 