The Penrose conjecture for initial data sets satisfying a $2$-convexity condition

Abstract

Let $(M^3,g,\mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon $\hat{I}\pounds$. We prove the Penrose conjecture, namely that $m_{\mathrm{ADM}}(g)\ge \sqrt{\frac{|\hat{I}\pounds |}{16\ddot{I}€}}$, under the assumptions of the dominant energy condition and the $2$-convexity condition that the sum of the two smallest eigenvalues of $\mathbf{k}$ is nonnegative. The main tool is the $\ddot{I}ƒ$-inverse mean curvature flow, together with a monotonicity formula developed in \cite{Dong26SigmaIMCF}.

v2: connected horizon assumption added; references added;