The Angular Momentum Penrose Inequality

We prove the Penrose inequality with angular momentum for asymptotically flat, axisymmetric vacuum initial data sets containing a stable marginally outer trapped surface. This inequality provides a lower bound for the ADM mass in terms of the area and angular momentum of the black hole horizon, with equality holding if and only if the initial data set corresponds to a slice of the Kerr spacetime. Our proof combines the Jang equation approach with the p-harmonic level set method. A key component of the analysis is a modified Hawking mass functional that incorporates angular momentum and exhibits monotonicity along the flow. We also establish the rigidity of the inequality using the Mars-Simon tensor.