  In this paper we describe the notion of an annular end of a Riemann surface
being of finite type with respect to some harmonic function and prove some
theoretical results relating the conformal structure of such an annular end to
the level sets of the harmonic function. We then apply these results to
understand and characterize properly immersed minimal surfaces in
$\mathbb{R}^3$ of finite total curvature, in terms of their intersections with
two nonparallel planes.
