The $\sigma$-inverse mean curvature flow and the generalized Penrose conjecture

Let $(M^3, g, \mathbf{k})$ be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let $m$ denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon $N\subset M$ satisfies $|N| \leq 16 \pi m^2$. In this paper, we establish this inequality for each connected component of $N$ in the special case where $\mathbf{k}$ is proportional to the metric $g$. Our approach is based on a new geometric evolution, which we call the $\sigma$-inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.