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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 190
Deformations of isotropic curves in C^3 and minimal surfaces in R^3 H. Gollek
We study properties of the natural parametrization of E. Vessiot and E. Study of isotropic curves in 3-dimensional complex space and their Weierstrass-representation formula. Demanding, that the parametrization of the given curve is the natural one, we arrive at some special case of this formula, the natural Weierstrass-representation. We show that the mapping, assigning to a meromorphic function f this isotropic curve is equivariant with respect to a certain group-homomorphism of Sl(2,C) onto O(3,C) and study some consequences of this fact for symmetries of minimal surfaces. The so-called Study invariant (or minimal curvature) of an isotropic curve resembles the classical notion of curvature of space curves. We use it here for the construction of `variations` and `deformations` of this curve. These are new isotropic curves that depend on the given curve and an arbitrary meromorphic function h. If the curve is given by the the natural Weierstrass-representation in terms of an other meromorphic function f, the variation yields a new representation formula of minimal surfaces, depending on two arbitrary meromorphic functions f and h and their derivatives only, not involving integrations.
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