COMPLETE EMBEDDED HARMONIC SURFACES IN R3

Abstract. Embedded minimal surfaces of finite total curvature in R3 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 R3 of compact Riemann sur-faces 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 2pi. 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. 1