We give formulae for minimal surfaces in ℝ3 deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ1. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.
Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C 3 and thus an algebraic minimal surface in R 3 . We derive simple formulae for the coordinate functions of such a null curve.
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