Complete Embedded Harmonic Surfaces In $\mathbb{R}^3$

TL;DR

This paper explores the classification and construction of complete embedded harmonic surfaces in three-dimensional space, focusing on their ends and total curvature, extending understanding beyond minimal surfaces.

Contribution

It provides a classification of harmonic ends with small total curvature, constructs examples with large total curvature, and explores surfaces with complex topology.

Findings

Classified embedded harmonic ends of small total curvature

Constructed examples of ends with arbitrarily large total curvature

Classified complete embedded harmonic surfaces of small total curvature

Abstract

Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. This paper is motivated by two outstanding features of such surfaces: They can have highly complicated ends, and they still have total Gauss curvature being a multiple of $2\pi$. This poses the double challenge to construct and classify examples of fixed total Gauss curvature. Our results include - a classification of embedded harmonic ends of small total curvature, - the construction of examples of embedded ends of arbitrarily large total curvature, - a classification of complete embedded harmonic surfaces of small total curvature in the spirit of the corresponding classification of minimal surfaces of small total curvature, and - the largely experimental construction of complete embedded harmonic surfaces with non-trivial topology that incorporate some of the new harmonic ends.