Topology of Cosmological Black Holes

Motivated by the question of how generic inflation is, I study the time-evolution of topological surfaces in an inhomogeneous cosmology with positive cosmological constant $\mathrm{\Lambda }$. If matter fields satisfy the Weak Energy Condition, non-spherical incompressible surfaces of least area are shown to expand at least exponentially, with rate $d \log A_{\rm min}/d\lambda \geq 8\pi G_N\Lambda$,underthemeancurvatureflowparametrizedby$\lambda$. With reasonable assumptions about the nature of singularities this restricts the topology of black holes: (a) no trapped surface or apparent horizon can be a non-spherical, incompressible surface, and (b) the interior of black holes cannot contain any such surface.