### Abstract:

In this dissertation, we study geometric inequalities for black holes, mainly the angular momentum-mass inequality and the angular momentum-mass-charge inequality. Firstly, we show how to reduce the general formulation of the angular momentum-mass inequality, for (non-maximal) axially symmetric initial data of the Einstein equations, to the known maximal case. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition. More importantly, we compute the scalar curvature formula for the deformation of initial data, which shows that the dominant energy condition holds in a weak sense. Through this procedure, we develop a geometrically motivated system of quasi-linear elliptic equations which is conjectured to admit a solution. The primary equation bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Secondly, in a similar sense, we show how to reduce the general formulation of the angular momentum-mass-charge inequality, for (non-maximal) axially symmetric initial data of the Einstein-Maxwell equations with zero magnetic field, to the known maximal case, whenever there exists a solution for the system of quasi-linear elliptic equations. Lastly, we combine these two results and the area-angular momentum inequality to show the lower bound of the area in terms of ADM mass, angular momentum, and charge for black holes under the same assumptions.