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Deformations of minimal surfaces of \(\mathbb{R}^3\) containing planar geodesics. (English) Zbl 0985.53004
Slovák, Jan (ed.) et al., Proceedings of the 18th winter school “Geometry and physics”, Srní, Czech Republic, January 10-17, 1998. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 59, 143-153 (1999).
The author begins by reviewing two ways to obtain representations for solutions of the minimal surface equation in \({\mathbb{R}}^3\). The first is the well-known Weierstrass formula. The second is the Björling representation formula, which is parameterized in terms of an analytic strip, i.e. a pair of curves \(( \alpha, \gamma)\) such that \(\langle \alpha', \gamma \rangle =0\) and \(|\gamma'|=1\). The Björling representation takes on a special form (which the author shows can be easily related to the Weierstrass data) when the curve \(\alpha\) is planar and the cuve \(\gamma\) is taken to be the normal to the plane containing \(\alpha\).
The natural parameter \(p\) (or Study parameter) and minimal curvature \(\kappa\) are discussed, where the natural parameter for non-planar minimal curves is obtained by normalizing the Hopf differential and the minimal curvature is given by, up to scalar, the square root of the Gauss map’s Schwarzian derivative. The author has a more in-depth exposition of these quantities in [Deformations of minimal curves in \(\mathbb{C}^3\), in: D. Wójcik (ed.), Proc. Warsaw 1995, 269-286 (1998; Zbl 0962.53007)].
After discussing a group homomorphism from \(Gl(2, {\mathbb{C}})\) to \(SO(3,{\mathbb{C}})\), it is stated that the natural parameter uniquely determines a minimal curve up to elements of \(SO(3, \mathbb{C})\) and translations. (loc. cit. and H. Pabel, Deformationen von Minimalflächen. in: O. Giering (ed.), Geometric und ihre Anwendungen, Carl Hanser Verlag, 107-139 (1994; Zbl 0966.53008), are the references given for a complete proof.) The fourth derivative \(\Phi_{pppp}\) of the minimal curve \(\Phi\) with respect to its natural parameter is expressed in terms of the lower derivatives. (Again see [the author, loc. cit.] for an exposition of the Frenet frame analog that can be developed using these parameters.)
These parameters, \(p\) and \(\kappa\), are then investigated in the context of the specialized Björling representation (where \(\alpha\) is a planar curve): both can be given in terms of the signed curvature of the planar curve \(\alpha\) and they satisfy a differential equation. A proposition states that, conversely, if for \(\Phi\) one has \(\kappa\) and \(p\) satisfying the differential equation then there is a planar curve \(\alpha\) such that \(\Phi\) can be obtained through the specialized Björling formula.
Deformations \(\Psi\) and \(\Delta\) of a given minimal curve \(\Phi\) that both have the same infinitesmal arclength are then constructed – the deformation also depends on the (free) choice of the function \(h\). These deformations are then used to define operators on the algebra of meromorphic functions, and the effects of parameter changes on the operators are examined before examining the deformations in the special context of the Björling formula. Examples of the deformations of the Björling formula, with nice graphics, conclude this paper.
For the entire collection see [Zbl 0913.00039].
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A04 Curves in Euclidean and related spaces
34L99 Ordinary differential operators