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In recent work, the notion of Double Convexity for a foliation of a conicalnull hypersurface was introduced to give a proof, if satisfied, of the NullPenrose Inequality. Double Convexity constrains the geometry of a MarginallyOuter Trapped Surface (MOTS), called a quasi-round MOTS. In the first part ofthis paper, for a class of strictly stable Weakly Isolated Horizons, we showthe existence of a unique foliation by quasi-round MOTS. In the second part, weshow that any subsequent space-time perturbation continues to admit aquasi-round MOTS. Finally, for perturbations of the quasi-round MOTS inSchwarzschild, we identify sufficient conditions on the asymptotics of anypast-pointing null hypersurface that yields the Null Penrose Inequality.

Quasi-Round MOTSs and Stability of the Schwarzschild Null Penrose Inequality