Abstract
Embedded minimal surfaces of finite total Gaussian curvature in are well understood as intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in of compact Riemann surfaces with finitely many punctures parameterized by meromorphic data. This paper is motivated by two outstanding features of such surfaces: they can have highly complicated ends, and they still have total curvature a multiple of 2π. This poses the dual challenge of constructing and classifying examples of fixed total 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 properly 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 nontrivial topology that incorporate some of the new harmonic ends.
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