Semi-discrete isothermic surfaces

Abstract

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory.

Channel surfaces in Lie sphere geometry

“…Remark 5.4 Theorem 4.7 applied to the particular case of curves recovers a result given in Burstall et al (2016): for any Ribaucour pair of curves we can choose a polarization such that it becomes a Darboux pair.…”

“…Burstall et al 2016;Hertrich-Jeromin 2003) Two curves form a Ribaucour pair if they envelop a circle congruence.…”

Transformations and singularities of polarized curves

“…In this section we define Darboux and Calapso transforms of polarized curves, show that our definitions agree with those of [6] and specify the goal of this paper: the study of the limiting behaviour of Darboux and Calapso transforms at points where the polarization has a pole of first or second order.…”

“…Namely, let ( c , Q) be a polarized curve with associated 1-form ω and orthogonal lift Ω. Using (6), it follows that withĉ p ∈ ĉ p the liftĉ = Γ p (λΩ)ĉ p of a λ-Darboux transform ĉ of ( c , Q) as defined in Def 2.1 is a parallel section of the connection Not surprisingly, the main advantage of the linearisation of the projective formalism is its linear structure. For example, integration of (3) shows that the primitives of an o(R n+2…”

Ribaucour coordinates

“…Recall that a semi-discrete curvature line net can be thought of as a sequence of curves, where subsequent curves form Ribaucour pairs, cf [13, Def 1.1] or [7,Sect 3]. As the construction of Cor 2.3 of a channel surface from a Ribaucour pair of curves involves the choice of a parallel normal field along one of the two curves, this construction may be iterated to "smoothen" a semi-discrete curvature line net, see Fig 3. reminiscent of the smoothing of fully discrete curvature line nets by using Dupin cyclides, cf [4]:…”