Abstract
This chapter gives a brief history of minimal immersions in Euclidean space. We present the calculation of the first variation of the area functional and we derive the Enneper–Weierstrass representation. Scherk’s surface is used to illustrate the problems that arise in integrating the Weierstrass forms. This integration problem is a simpler version of the monodromy problem encountered later in finding examples of CMC 1 immersions in hyperbolic geometry. We present results on complete minimal immersions with finite total curvature, which will be used in Chapter 14 to characterize minimal immersions in Euclidean space that smoothly extend to compact Will more immersions into Mobius space. The final section on minimal curves applies the method of moving frames to the nonintuitive setting of holomorphic curves in C3 whose tangent vector is nonzero and isotropic at every point.
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