  We explore the problem of localization in topological and non-topological
nearly-flat subbands derived from the lowest Landau level, in the presence of
quenched disorder and short-range interactions. We consider two models: a
suitably engineered periodic potential, and randomly distributed point-like
impurities. We perform numerical exact diagonalization on a torus geometry and
use the mean level spacing ratio $\langle r \rangle$ as a diagnostic of
ergodicity. For topological subbands, we find there is no ergodicity breaking
in both the one and two dimensional thermodynamic limits. For non-topological
subbands, in constrast, we find evidence of an ergodicity breaking transition
at finite disorder strength in the one-dimensional thermodynamic limit.
Intriguingly, indications of similar behavior in the two-dimensional
thermodynamic limit are found, as well. This constitutes a novel,
$\textit{continuum}$ setting for the study of the many-body localization
transition in one and two dimensions.
