Finite type annular ends for harmonic functions

TL;DR

This paper introduces the concept of finite type annular ends for harmonic functions on Riemann surfaces and explores their relation to minimal surfaces in three-dimensional space, providing new characterizations.

Contribution

It defines finite type annular ends for harmonic functions and links their conformal structure to minimal surface intersections with planes, advancing geometric analysis.

Findings

Characterization of finite type annular ends via level sets

Relation between conformal structure and harmonic function behavior

Application to classifying minimal surfaces with finite total curvature

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. 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.