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

Let $(M^3, g, \mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with outermost past apparent horizon.

We prove the Penrose conjecture for each connected component of the boundary, assuming 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 $\sigma $-inverse mean curvature flow and a monotonicity formula developed in \cite{Dong26SigmaIMCF}.