  On a flat surface, the Landau operator, or quantum Hall Hamiltonian, has
spectrum a discrete set of infinitely degenerate Landau levels. We consider
surfaces with asymptotically constant curvature away from a possibly
non-compact submanifold, the helicoid being our main example. The Landau levels
remain isolated, provided the spectrum is considered in an appropriate Hilbert
module over the Roe algebra of the surface delocalized away from the
submanifold. Delocalized coarse indices may then be assigned to them. As an
application, we prove that Landau operators on helical surfaces have no
spectral gaps above the lowest Landau level.
