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Comments on Penrose inequality with angular momentum for outermost apparent horizons

In a recent work we have proved a weaker version of the Penrose inequality with angular momentum, in axially symmetric space-times, for a compact and connected minimal surface. In this previous work we use the monotonicity of Geroch energy on 2-surfaces along the inverse mean curvature flow and we obtain a lower bound for the ADM mass in terms of the area, the angular momentun and a particular measure of size of the minimal surface. In the present work, using similar techniques and the same measure of size, we extend and improve the previous result for a compact and connected outermost apparent horizon. For this case we use the monotonicity of Hawking energy, instead of Geroch energy, along the inverse mean curvature flow, and assume different conditions on the extrinsic curvature. This type of relations constitutes an important test to evaluate the cosmic censorship conjecture.