Abstract
We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached, all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index frac{1}{2} or 1 is approached along a curvature line.
Highlights
The importance of transformations becomes even more apparent, as articulated in [3]: “In this setting, discrete surfaces appear as two-dimensional layers of multidimensional discrete nets, and their transformations correspond to shifts in the transversal lattice directions
In analogy to the transformation theory of smooth isothermic surfaces, the authors develop a notion of Christoffel, Darboux and Calapso transformations of polarized curves, that is, smooth curves equipped with a nowhere zero quadratic differential, called a polarization
We are interested in the class of smooth isothermic surfaces, classically characterized by the local existence of conformal curvature line coordinates, away from umbilics
Summary
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. This paper is devoted to the study of the limiting behaviour of the Darboux and Calapso transforms at a in the case that c can be extended regularly to some (a − , b), but the polarization has a pole of first or second order at a in the sense of Definition 2 A quadratic differential Q : (a, b) t → Qt = Q(t) d t2 or a Lie algebravalued 1-form ψ : (a, b) t → ψt = ψ(t) d t on (a, b) has a pole of order k ∈ N at a if the function t → Q(t)(a − t)k or t → ψ(t)(a − t)k, respectively, has a smooth extension to (a − , b) for some > 0 with nonzero value at a.
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