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Minimal Immersions in Euclidean Space

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Surfaces in Classical Geometries

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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 Möbius space. The final section on minimal curves applies the method of moving frames to the nonintuitive setting of holomorphic curves in C 3 whose tangent vector is nonzero and isotropic at every point.

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Authors and Affiliations

  1. Department of Mathematics, Washington University, St. Louis, MO, USA

    Gary R. Jensen

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy

    Emilio Musso

  3. Dipartimento di Matematica e Informatica, Università di Parma, Parma, Italy

    Lorenzo Nicolodi

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  1. Gary R. Jensen
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  2. Emilio Musso
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  3. Lorenzo Nicolodi
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Jensen, G.R., Musso, E., Nicolodi, L. (2016). Minimal Immersions in Euclidean Space. In: Surfaces in Classical Geometries. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27076-0_8

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