  We study how the stability of the fractional quantum Hall effect (FQHE) is
influenced by the geometry of band structure in lattice Chern insulators. We
consider the Hofstadter model, which converges to continuum Landau levels in
the limit of small flux per plaquette. This gives us a degree of analytic
control not possible in generic lattice models, and we are able to obtain
analytic expressions for the relevant geometric criteria. These may be
differentiated by whether they converge exponentially or polynomially to the
continuum limit. We demonstrate that the latter criteria have a dominant effect
on the physics of interacting particles in Hofstadter bands in this low flux
density regime. In particular, we show that the many-body gap depends
monotonically on a band-geometric criterion related to the trace of the
Fubini-Study metric.
