On the asymptotics of 3+1D cosmologies with bounded scalar potential and isometry group forming 2-dimensional orbits

We study the onset of inflation in 3+1 dimensional cosmologies with an inflationary potential $U$ satisfying $0 < \Lambda_1 \leq U \leq \Lambda_2$, matter satisfying the dominant and strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are orbits under an isometry group. Assuming an initial Cauchy slice with positive mean curvature everywhere, we show, via mean curvature flow, that there exists a family of spatial slices parameterized by $\lambda$, whose volume grows between the flat slicings in de Sitter spaces with cosmological constants $\Lambda_1$ and $\Lambda_2$. In particular, inflationary expansion indeed occurs in this setting with inhomogeneous initial conditions. Finally, we apply this "inflationary time coordinate" $\lambda$ to study asymptotics of the variation in the metric, the average stress-energy tensor, and the dynamics of an inflaton field on a spatial slice.