The generalized Penrose conjecture for asymptotically flat initial data sets

Let $(M^3,g,\mathbf{k})$ be a complete connected asymptotically flat initial data set satisfying the dominant energy condition. Choose an end with ADM mass $m$, and let $\mathcal{S}$ be the collection of compact $C^2$ boundaries of open sets that are bounded in the chosen end and contain all other ends. A surface $\Sigma\in\mathcal{S}$ is generalized trapped when

$$H_{\Sigma}\leq |\operatorname{tr}_{\Sigma}\mathbf{k}|.$$

Such a surface is enclosed by an outermost generalized apparent horizon $\widetilde{\Sigma}$, and write $A$ for its area. The generalized Penrose conjecture. If $\Sigma\in\mathcal{S}$ is a generalized trapped surface, then

$$m\geq \sqrt{\frac{A}{16\pi}}.$$

Equivalently, the area of the outermost generalized apparent horizon is at most $16\pi m^2$. This conjecture generalizes the Penrose inequality from time-symmetric data to general initial data satisfying the dominant energy condition; its resolution is a central problem in general relativity and geometric analysis.