Finite type annular ends for harmonic functions

Abstract

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. As an application of these results, we obtain important information on the conformal type of any properly immersed minimal surface M in $$\mathbb {R}^3$$ with compact boundary and which intersects some plane transversely in a finite number of arcs; in particular, such an M is a parabolic Riemann surface. This information is applied by Meeks III and Perez (Embedded minimal surfaces of finite topology, 2015) to classify the asymptotic behavior of annular ends of a complete embedded minimal annulus with compact boundary in terms of the flux vector along its boundary. In the present paper, we apply this information 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.