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Abstract: We consider compact connected minimal surfaces, with a pair of boundarycurves not necessarily convex in distinct planes, that have least-areaamongst all orientable surfaces with the same boundary. When the planescontaining these two boundary curves are either parallel or sufficiently closeto parallel, and when the boundary curves themselves are sufficiently close toeach other, we draw specific conclusions about the geometry and topology of thesurfaces. We also strength the following result: Let $M$ be any compact minimalannulus with two planar boundary curves of diameters $d 1$ and $d 2$ inparallel planes $P 1$ and $P 2$; if the distance between $P 1$ and $P 2$ is$h$, then the inequality $h \leq {3-2}\max\{d 1,d 2\}$ is satisfied. Westrength it by removing the assumption that $M$ is an annulus and by showingthat the stronger conclusion $h \leq \max\{d 1,d 2\}$ holds. We also include asimilar result for nonminimal constant mean curvature surfaces.

Autor: Wayne Rossman


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