  In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for
a 3-dimensional initial data set $(M,g,K)$ with mean curvature $\tau\le0$ in a
4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable
assumptions) $M$ has a stable spherical MOTS $\Sigma$ which achieves an upper
bound for the area. He proved that each leaf is a round 2-sphere with the same
constant Gaussian curvature. Here, we improve his result by dropping the
hypothesis on the mean curvature of $M$ and showing that each leaf is a minimal
surface. We show that in an outer neighborhood $U$ of $\Sigma$, the metric $g$
splits as the product $([0,\varepsilon)\times\Sigma,dt^2+g_0)$, where
$(\Sigma,g_0)$ is a round 2-sphere. In fact, we prove that the outer
neighborhood $U$ can be isometrically embedded into the 4-dimensional Nariai
spacetime $(\overline N,\overline h)$ as a spacelike hypersurface so that
$g|_U$ is the induced metric from $\overline N$ and $K|_U$ is the second
fundamental form of $U$ in $\overline N$. This completely classifies the local
geometry of such initial data sets.
