# Transformations and singularities of polarized curves

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## 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.

## Keywords

Isothermic surfaces Singular umbilics Polarized curve Darboux transformation Calapso transformation## Mathematics Subject Classification

53A30 37K35 37K40 58K45## 1 Introduction

Transformations of surfaces play a central role in our present understanding of smooth and discrete differential geometry. Not only do they allow the construction of new surfaces of a given class from existing ones, but their existence reveals a great deal about the underlying (integrable) structure of the corresponding classes of surfaces, c.f. [5]. For example, given a pseudospherical surface (\(K=-\,1\)), the Bäcklund transformation yields a two-parameter family of new pseudospherical surfaces by solving an integrable, first-order partial differential equation (see [11, Sect. 120]). In [22], these Bäcklund transformations are shown to be generators of an infinite-dimensional transformation group of the space of pseudospherical surfaces and the relations among these generators are given by a permutability theorem discovered by Bianchi [1, Sect. 257] in the nineteenth century. In particular, it is shown that pseudospherical surfaces fall into the class of integrable systems. Similar developments have been achieved for other classes of surfaces such as minimal surfaces, constant mean curvature surfaces, constant Gaussian curvature surfaces (c.f. [2]) and curved flats in symmetric spaces such as Darboux pairs of isothermic surfaces, c.f. [7, 9, 12].

In the discrete theory, the importance of transformations becomes even more apparent, as articulated in [3] (see also [4]): “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. A characteristic feature of the theory is that all lattice directions are on equal footing with respect to the defining geometric properties. Due to this symmetry, discrete surfaces and their transformations become indistinguishable.”

The interplay between aspects of discrete and smooth differential geometry was explored in [6] with the study of semi-discrete isothermic surfaces, introduced in [18]. In analogy to the transformation theory of smooth isothermic surfaces (see [15, Chap. 8.6], [5] or [8]), 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. They then show that semi-discrete isothermic surfaces are sequences of Darboux transforms of polarized curves. In line with the ideas of discrete differential geometry, permutability theorems of the transformations of polarized curves are shown to yield a corresponding transformation theory for semi-discrete isothermic surfaces.

We are interested in the class of smooth isothermic surfaces, classically characterized by the local existence of conformal curvature line coordinates, away from umbilics. For these surfaces, the transformation theory is defined only locally, that is, on simply connected surface patches on which regular nets of conformal curvature line coordinates exist (c.f. [5, 15, 20]). A global transformation theory is still missing. In particular, it seems necessary to reconsider the definition of an isothermic surface. The classical definition makes no restriction whatsoever on umbilic points. But certain configurations of umbilics are an obstacle for a global definition of the transformations (see [15, Sect. 5.2.20]). One candidate for an alternative definition of an isothermic surface is to require the existence of a globally defined holomorphic quadratic differential whose trajectories (see [21, Sect. 5.5]) agree with the curvature lines of the surface on the complement of its umbilic set. But due to the Poincaré–Hopf theorem [16], this would exclude all surfaces homeomorphic to a sphere, for example the ellipsoid, c.f. [20]. If we merely require the existence of a meromorphic quadratic differential, that includes topological spheres. Moreover, according to the local Carathéodory conjecture, the poles of such a meromorphic quadratic differential are at most of order two (see [14]).

This paper makes a first step towards a global transformation theory of isothermic surfaces for which an underlying meromorphic quadratic differential exists. A curvature line of an isothermic surface together with the restriction of such a meromorphic quadratic differential yields a polarized curve in the sense of [6, Sect. 2]. Moreover, the transformations for isothermic surfaces descend to the corresponding transformations of the polarized curvature lines. Thus, in order to understand how the transforms of an isothermic surface behave when a pole of the underlying meromorphic quadratic differential is approached along a curvature line, we investigate how Darboux and Calapso transforms of polarized curves with singular polarizations behave when the singularity is approached. In particular, we investigate the limiting behaviour of Darboux and Calapso transforms of polarized curves where the polarization has a pole of first or second order. According to the local Carathéodory conjecture, these are the only poles that can occur on a curvature line of an isothermic surface.

In Sect. 2, we give Möbius geometric definitions of the Darboux and Calapso transformations of polarized curves in the conformal *n*-dimensional sphere \(S^n\). Our definitions are formulated with a projective model of Möbius geometry: we identify \(S^n\) with the projectivization \(\mathbb {P}(\mathcal {L}^{n+1})\) of the light cone \(\mathcal {L}^{n+1}\) in \((n+2)\)-dimensional Minkowski space. The Möbius group can then be identified with the projective Lorentz group \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \). Curves are described as maps into the projective light cone. For computations and in order to prove that our definitions agree with those of [6], we relate the projective model to its linearization based on the action of \(O\left( \mathbb {R}^{n+2}_1\right) \) on \(\mathcal {L}^{n+1}\), where curves are described via their light cone lifts. We then give a definition of a polarization with a pole of first or second order.

In Sect. 4, we use these results to show that when the polarization of a polarized curve has a pole of first order, every Calapso transform converges to some point and all Darboux transforms converge to the original curve.

In Sect. 5, we consider the case of a polarization with a pole of second order. We cannot apply the results of Sect. 3 here directly because the 1-forms \(\lambda \omega \) associated with the polarized curve have poles of second order. We first have to do a singular gauge transformation to transform the 1-forms with poles of second order to 1-forms with a pole of first order. The behaviour of the Darboux and Calapso transforms in this case is more diverse. A generic Darboux transform still converges to the original curve as the singularity is approached, but there are Darboux transforms which do not converge. Moreover, the Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter \(\lambda \).

## 2 Darboux and Calapso transforms 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.

*n*-sphere \(S^n\) with the projectivization \(\mathbb {P}(\mathcal {L}^{n+1})\) of the light cone in \((n+2)\)-dimensional Minkowski space. In this way, the action of the Möbius group on \(S^n\) can be identified with the action of the projective Lorentz group \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) on \(\mathbb {P}(\mathcal {L}^{n+1})\). We view \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) as a subgroup of the group of projective transformations of \(\mathbb {P}\left( \mathbb {R}^{n+2}_1\right) \), which in turn is a submanifold of \(\mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \). Taking the linear span \(\langle \cdot \rangle \) of elements of \({\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \) then provides a diffeomorphism from the group \(O^+\left( \mathbb {R}^{n+2}_1\right) \subset {\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \) of orthochronous Lorentz transformations to \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \). Similarly, we view the Lie algebra \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \) as a subspace of the tangent space \({\textit{End}}\left( \mathbb {R}^{n+2}_1\right) /\mathbb {R}\, \mathrm{id}\) of \(\mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \) at the identity. The differential of \(\langle \cdot \rangle \) at the identity then restricts to an isomorphism of Lie algebras

*a*,

*b*) into \(\mathbb {P}(\mathcal {L}^{n+1})\), where the notation \(\langle c\rangle \) indicates that such a map may also be described by pointwise taking the linear span of a light cone lift \(c:(a,b)\rightarrow \mathcal {L}^{n+1}\) of \(\langle c\rangle \). For simplicity, we always assume \(\langle c\rangle \) to be smooth. A polarized curve \((\langle c\rangle ,Q)\) in \(S^n\) is a regular curve \(\langle c\rangle \) in \(S^n\) together with a nowhere zero quadratic differential

*Q*on (

*a*,

*b*). To a polarized curve \((\langle c\rangle ,Q)\), we associate the \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \)-valued 1-form \(\omega \) given by

*c*is any light cone lift of \(\langle c\rangle \) and the function \(\mathfrak {Q}\) is related to

*Q*via that lift

*c*by

*c*of \(\langle c\rangle \).

*G*be a Lie group with Lie algebra \(\mathfrak {g}\) and \(\psi \) a \(\mathfrak {g}\)-valued 1-form on (

*a*,

*b*). Then, for any \(p\in (a,b)\), denote by

*G*and thus satisfies

### Definition 1

Let \((\langle c\rangle ,Q)\) be a polarized curve in \(S^n\) with associated 1-form \(\omega \) given by (2).

*Darboux transform*of \((\langle c\rangle ,Q)\) with initial point \(\langle {\hat{c}}(p)\rangle =\langle {\hat{c}}_p\rangle \).

*Calapso transform of*\((\langle c\rangle ,Q)\)

*normalized at*

*p*.

Any \(\lambda \)-Calapso transform of a polarized curve \((\langle c\rangle ,Q)\) is an immersion. Similarly, any \(\lambda \)-Darboux transform with initial point \(\langle {\hat{c}}_p\rangle \) not lying on the image of \(\langle c_{\lambda ,p}\rangle \) immerses.^{1} One may turn the immersed transforms of \((\langle c\rangle ,Q)\) into polarized curves themselves by equipping them with some polarization, for example again *Q*. This is useful in the study of repeated transforms of polarized curves, as it is done in [6]. However, in this work we do not study repeated transforms and thus, for simplicity, we consider the transforms merely as curves and not as polarized curves.

We remark that due to (4), two \(\lambda \)-Calapso transforms normalized at *p* and *q*, respectively, differ by a Möbius transformation. Moreover, the Darboux and Calapso transformations of a polarized curve \((\langle c\rangle ,Q)\) in \(S^n\) are invariant under Möbius transformations of \(S^n\) and reparametrizations of \(\langle c\rangle \).

This paper is devoted to the study of the limiting behaviour of the Darboux and Calapso transforms at *a* in the case that \(\langle c\rangle \) can be extended regularly to some \((a-\epsilon ,b)\), but the polarization has a pole of first or second order at *a* in the sense of

### Definition 2

A quadratic differential \(Q:(a,b)\ni t\mapsto Q_t=Q(t){{\mathrm{d}}}t^2\) or a Lie algebra-valued 1-form \(\psi :(a,b)\ni t\mapsto \psi _t=\psi (t){{\mathrm{d}}}t\) on (*a*, *b*) has a pole of order \(k\in {\mathbb {N}}\) at *a* if the function \(t\mapsto Q(t)(a-t)^k\) or \(t\mapsto \psi (t)(a-t)^k\), respectively, has a smooth extension to \((a-\epsilon ,b)\) for some \(\epsilon >0\) with nonzero value at *a*.

If \(\langle c\rangle \) has a smooth extension to \((a-\epsilon ,b)\), then the polarization *Q* has a pole of first or second order at *a* if and only if the 1-form \(\omega \) associated with \((\langle c\rangle ,Q)\) has a pole of first or second order at *a*.

*a*,

*b*) to \((\tilde{a},\tilde{b})\) which extend to diffeomorphisms of some larger interval \((a-\epsilon ,b)\) to \((\tilde{a}-\tilde{\epsilon },\tilde{b})\) with \(\epsilon ,\tilde{\epsilon }>0\), for example the translation

*b*) and \(\varPsi \) its

*orthogonal lift*defined by \(\varPsi +\mathbb {R}~\mathrm{id}=\psi \) via the isomorphism (1). Then, the \(O^+\left( \mathbb {R}^{n+2}_1\right) \)-valued primitives \(\varGamma _p(\varPsi )\) satisfy

*h*on (0,

*b*), we have

^{2}of \((\langle c\rangle ,Q)\) in the sense of [6, Definition 2.5]. Conversely, any parallel section \({\hat{c}}\) of \(\frac{D^\lambda }{{{\mathrm{d}}}t}\) yields a \(\lambda \)-Darboux transform \(\langle {\hat{c}}\rangle \) according to our Definition 1.

Similarly, the \(\lambda \)-Calapso transform normalized at some \(p\in (0,b)\) up to Möbius transformation corresponds to the \(\left( -\tfrac{\lambda }{2}\right) \)-Calapso transform defined in [6, Definition 3.4] because, up to Möbius transformation, their Calapso transformation \(T^{-\lambda /2}\) is precisely \(\varGamma _p(\lambda \varOmega )\).

*b*) satisfy the useful integral equation in \({\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \simeq \mathbb {R}^{(n+2)^2}\)

*b*) have a pole, it may happen that the \(O^+\left( \mathbb {R}^{n+2}_1\right) \)-valued primitives \(\varGamma _p(\varPsi )\) tend towards infinity as the pole is approached, while the corresponding \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \)-valued primitives \(\varGamma _p(\psi )\) have limits in \(\mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \supset \mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \). If this is the case for the 1-form \(\omega \) associated with a polarized curve and its orthogonal lift \(\varOmega \), then the lifts \({\hat{c}}=\varGamma ^{p}(\lambda \varOmega ){\hat{c}}_p\) and \(c_{\lambda ,p}=\varGamma _p(\lambda \varOmega )c\) diverge as the singularity is approached, while the corresponding maps \(\langle {\hat{c}}\rangle \) and \(\langle c_{\lambda ,p}\rangle \) converge. In such a situation, a lift-independent formalism is clearly beneficial.

## 3 Main tools

In this section, we study the limiting behaviour at 0 of the primitives \(\varGamma _{p}(\psi )\) of a certain class of \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \)-valued 1-forms \(\psi \) on (0, *b*) with a pole of first order at 0. In Sect. 3.1, we restrict to 1-forms of a particularly simple form, which we call *pure pole forms*. For those, the primitives and their limiting behaviour at 0 can be computed explicitly. In Sect. 3.2, we then relate the limiting behaviour of primitives of the more general pole forms to that of the pure pole forms. In view of our definition of the transformations of a polarized curve via the primitives of multiples \(\lambda \omega \) of the associated 1-form \(\omega \) (Definition 1), these tools will provide the means to investigate the limiting behaviour of the transforms of a singular polarized curve in Sects. 4 and 5.

### 3.1 Primitives of pure pole forms

*b*) a

*pure pole form*if it is of the form

In addition to the quadratic form \(\Vert {\cdot }\Vert ^2\) induced by the indefinite Minkowski inner product Open image in new window , we introduce the standard, Euclidean, positive-definite norm \(|{\cdot }|\) on \(\mathbb {R}^{n+2}\) and \(\mathbb {R}^{(n+2)^2}\). On \({\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \simeq \mathbb {R}^{(n+2)^2}\), the norm \(|{\cdot }|\) is submultiplicative and satisfies \(| A^*|=|A|\) for \(A^*\) the adjoint of \(A\in {\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \) with respect to the *Minkowski* inner product on \(\mathbb {R}^{n+2}_1\). On \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \), we define \(|{\cdot }|\) by the push forward of \(|{\cdot }|\) restricted to \(\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \subset \mathbb {R}^{(n+2)^2}\) via the isomorphism (1). Furthermore, for \(w\in \mathbb {R}^{n+2}_1\), we denote by \(w^*\) the linear map Open image in new window .

### Proposition 1

*b*) and \(\varvec{\varXi }\) its orthogonal lift. Then, there is a constant \(B\in \mathbb {R}\) such that

- 1.if \(\xi \) is Minkowski, thenwhere \(v_\pm \in \langle v,w\rangle \cap \mathcal {L}^{n+1}\) are eigenvectors of \(v\wedge w\) with eigenvalues \(\pm \zeta \), \(\zeta >0\);$$\begin{aligned} \forall p\in (0,b)~\forall t\in (0,p]:\,&\left| \left( \frac{t}{p}\right) ^\zeta \varGamma _{p}^{t}(\varvec{\varXi })\right| < B, \end{aligned}$$(8)
- 2.if \(\xi \) is spacelike, thenbut \(\varGamma _p(\xi )\) does not converge as 0 is approached as it is a rotation in the plane \(\langle v,w\rangle \) with speed increasing towards infinity;$$\begin{aligned} \forall p,t\in (0,b):\,\left| \varGamma _{p}^{t}(\varvec{\varXi })\right| < B,\quad \end{aligned}$$(10)
- 3.if \(\xi \) is degenerate, choose \(v_0\in \langle v,w\rangle \cap \mathcal {L}^{n+1}\) and \(\tilde{w}\in \langle v,w\rangle \) such that \(v\wedge w=v_0\wedge \tilde{w}\). Then,$$\begin{aligned} \forall p,t\in (0,b):\,&\left| \frac{1}{\ell (p,t)}\varGamma _{p}^{t}(\varvec{\varXi })\right| <B, \end{aligned}$$(11)where \(\ell :(0,b)^2\rightarrow \mathbb {R}\) is the symmetric function$$\begin{aligned} \forall p\in (0,b):\,&\lim _{t\rightarrow 0}\frac{1}{\ell (p,t)}\varGamma _{p}^{t}(\varvec{\varXi })=-\frac{\Vert \tilde{w}\Vert ^2}{2} v_0v_0^*, \end{aligned}$$(12)$$\begin{aligned} \ell :(p,t)\mapsto 1+\left( \ln \left( \frac{t}{p}\right) \right) ^2. \end{aligned}$$(13)

### Proof

- 1.For \(\langle v,w\rangle \) Minkowski, we may choose \(v_\pm \in \mathcal {L}^{n+1}\) such that Open image in new window and \(v\wedge w=v_-\wedge v_+\). Then, \(v_\pm \) are eigenvectors of \(v\wedge w\) with eigenvalues Open image in new window and \(\langle v_+,v_-\rangle ^\perp =ker(v\wedge w)\). Using thatfor all \(s\in \mathbb {R}\) and \(k\in {\mathbb {N}}\), the exponential computes to$$\begin{aligned} (s~v_-\wedge v_+)^k=(s\zeta )^k~\left( \frac{v_+v_-^*}{\zeta }\right) ^k+(-\,s\zeta )^k\left( \frac{v_-v_+^*}{\zeta }\right) ^k=(s\zeta )^k\frac{v_+v_-^*}{\zeta }+(-\,s\zeta )^k\frac{v_-v_+^*}{\zeta }\end{aligned}$$Therefore,$$\begin{aligned} \varGamma _{p}^{t}(\varvec{\varXi })=\hbox {e}^{-\ln \left( \frac{t}{p}\right) v\wedge w}=\left( \mathrm{id}-\frac{v_+v_-^*+v_-v_+^*}{\zeta }\right) +\left( \frac{t}{p}\right) ^{-\zeta }~\frac{v_+v_-^*}{\zeta }+\left( \frac{t}{p}\right) ^{\zeta }~\frac{v_-v_+^*}{\zeta }. \end{aligned}$$(15)Since \(\zeta >0\), taking the limit \(t\rightarrow 0\) yields (9). Since \(|t/p|\le 1\) for all \(p\in (0,b)\) and \(t\in (0,p]\), there is a constant \(B\in \mathbb {R}\) such that (8) holds.$$\begin{aligned} \left( \frac{t}{p}\right) ^{\zeta }\varGamma _{p}^{t}(\varvec{\varXi })=\left( \frac{t}{p}\right) ^{\zeta }\left( \mathrm{id}-\frac{v_+v_-^*+v_-v_+^*}{\zeta }\right) +\frac{v_+v_-^*}{\zeta }+\left( \frac{t}{p}\right) ^{2\zeta }~\frac{v_-v_+^*}{\zeta }. \end{aligned}$$
- 2.
For \(\langle v,w\rangle \) spacelike, \(\varGamma _{p}^{t}(\varvec{\varXi })\) takes values in a one-parameter subgroup \(O(\mathbb {R}^2)\subset O^+\left( \mathbb {R}^{n+2}_1\right) \) of Euclidean rotations. Since \(O(\mathbb {R}^2)\) is compact, \(\varGamma _{p}^{t}(\varvec{\varXi })\) is bounded for all \(p,t\in (0,b)\) such that (10) holds for some \(B\in \mathbb {R}\). To see that \(\varGamma _p(\xi )\) does not have a limit at 0, denote by \(i\zeta \) a nonzero eigenvalue of \(v\wedge w\). Under a Lie group isomorphism \(O(\mathbb {R}^2)\rightarrow \mathbb {R}~\hbox {mod}~2\pi \), the primitive \(\varGamma _{p}^{t}(\varvec{\varXi })\) gets mapped to \(\pm \ln \left( \frac{t}{p}|\zeta |\right) ~\hbox {mod}~2\pi \) which does not have a limit for any \(p\in (0,b)\) as

*t*tends to 0. Hence, also \(\varGamma _{p}^{t}(\varvec{\varXi })\) does not have a limit for any \(p\in (0,b)\) as*t*tends to 0. - 3.Now let \(\xi \) be degenerate with \(v_0\in \langle v,w\rangle \cap \mathcal {L}^{n+1}\) and \(\tilde{w}\in \langle v,w\rangle \) such that \(v\wedge w=v_0\wedge \tilde{w}\). Then, \((v\wedge w)^2=-\Vert \tilde{w}\Vert ^2v_0v_0^*\) and \((v\wedge w)^k=0\) for all \(k\ge 3\). With \(\ell \) as in (13), we thus getTaking the limit \(t\rightarrow 0\) yields (12). The bound (11) follows from the boundedness of \(\frac{1}{1+x^2}\), \(\frac{x}{1+x^2}\), \(\frac{x^2}{1+x^2}\) for all \(x\in \mathbb {R}\).$$\begin{aligned} \frac{1}{\ell (p,t)}\varGamma _{p}^{t}(\varvec{\varXi })= \frac{\mathrm{id}-\ln \left( \frac{t}{p}\right) ~v\wedge w-\frac{\left( \ln \left( \frac{t}{p}\right) \right) ^2\Vert \tilde{w}\Vert ^2}{2}v_0v_0^*}{1+\left( \ln \left( \frac{t}{p}\right) \right) ^2}. \end{aligned}$$(16)

The one-parameter families \((\varGamma _{p}^{t}(\xi ))_{t\in (0,b)}\) have simple Euclidean interpretations: For Minkowski \(\xi \), stereographically project \(S^n\backslash \{ \langle v_+\rangle \}\) to \(\mathbb {R}^n\). Then, \((\varGamma _{p}^{t}(\xi ))_{t\in (0,b)}\) acts as a family of similarities with centre \(\langle v_-\rangle \) of \(\mathbb {R}^n\) and a *t*-dependent scale factor that tends towards infinity as \(t\rightarrow 0\). The limiting map \(\langle v_+v_-^*\rangle \) maps all points to the point at infinity, \(\langle v_+\rangle \), except the centre, \(\langle v_-\rangle \), where \(\langle v_+v_-^*\rangle \) is not defined.

If \(\xi \) is spacelike, \((\varGamma _{p}^{t}(\xi ))_{t\in (0,b)}\) takes values in a one-parameter subgroup \(O(\mathbb {R}^2)\subset O^+\left( \mathbb {R}^{n+2}_1\right) \) of Euclidean rotations. As *t* approaches 0, the rotation speed increases towards infinity.

For degenerate \(\xi \) and \(v_0\in \langle v,w\rangle \cap \mathcal {L}^{n+1}\), stereographically project \(S^n\backslash \{\langle v_0\rangle \}\) to \(\mathbb {R}^n\). Then, \((\varGamma _{p}^{t}(\xi ))_{t\in (0,b)}\) acts as a family of translations of \(\mathbb {R}^n\) by a vector of constant direction and *t*-dependent length which tends towards infinity as \(t\rightarrow 0\). The limiting map \(\langle v_0v_0^*\rangle \) maps all points to the point at infinity, \(\langle v_0\rangle \), except \(\langle v_0\rangle \) itself, where \(\langle v_0v_0^*\rangle \) is not defined.

These limits are not elements of \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \). Instead, they lie on the closure of \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) in \(\mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \). In fact, it can be shown (c.f. [13]) that the closure of \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) in \(\mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \) is compact and precisely the union of \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) and the set \(\{\langle vw^*\rangle ~|~v,w\in \mathcal {L}^{n+1}\}\subset \mathbb {P}{\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \). In contrast, \(O^+\left( \mathbb {R}^{n+2}_1\right) \) is closed in \({\textit{End}}\left( \mathbb {R}^{n+2}_1\right) \), but not compact. The limits (9) and (12) can thus not be computed in \(O^+\left( \mathbb {R}^{n+2}_1\right) \), which demonstrates the advantage of the projective model.

### 3.2 Primitives of pole forms

In this section, we define pole forms and relate the limiting behaviour of their primitives at 0 to that of pure pole forms.

### Definition 3

A \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \)-valued 1-form \(\psi \) on (0, *b*) is a *pole form* if \(\psi -\xi \) is bounded with respect to the Euclidean norm \(|{\cdot }|\) for a pure pole form \(\xi \) on (0, *b*).

For a pole form \(\psi \), the pure pole form \(\xi \) for which \(\psi -\xi \) is bounded is clearly unique. Therefore, we may define

### Definition 4

Let \(\psi \) be a pole form on (0, *b*) with \(\psi -\xi \) bounded for the pure pole form \(\xi \). Then, \(\psi \) is called *Minkowski, spacelike* or *degenerate* according to whether \(\xi \) has that property. For Minkowski \(\psi \), denote by \(\zeta \) the positive eigenvalue of \(v\wedge w\). If \(\psi \) is Minkowski with \(\zeta <1\), spacelike or degenerate, we say that \(\psi \) is *of the first kind*. Otherwise, it is *of the second kind*.

We remark that these properties of a pole form are invariant under those diffeomorphisms of (0, *b*) to \((\tilde{a},\tilde{b})\) which extend to diffeomorphisms of \((-\,\epsilon ,b)\) to \((\tilde{a}-\tilde{\epsilon },\tilde{b})\) for some \(\epsilon ,\tilde{\epsilon }>0\) (c.f. [13]).

Different techniques are necessary to investigate the limiting behaviour of primitives of pole forms of the first and of the second kind. We start with those of the first kind.

### Lemma 1

*b*), and let \(\chi \) be a bounded, continuous map from (0,

*b*) to \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \). Then, the family \(\left( \varGamma _{p}(\xi )\chi \varGamma ^{p}(\xi )\right) _{p\in (0,b)}\) of maps from (0,

*b*) to \(\mathfrak {p}\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \) satisfies

*b*) that satisfies

### Proof

If \(\xi \) is spacelike, again by Proposition 1, \(\varGamma _{p}(\varvec{\varXi })\) is bounded on (0, *b*). Therefore, \(\varGamma _{p}(\xi )\chi \varGamma ^{p}(\xi )\) is bounded on (0, *b*). Thus, we can choose \(\mathcal {B}_{p}\) to be the same constant for all \(p\in (0,b)\). Clearly, a constant \(B_{p}\) is integrable and satisfies (17) and (18).

*t*reads

*C*such that

*b*) and the product \(t\,\mathcal {B}_{p}(t)\) converges to zero as

*t*tends towards

*b*. \(\square \)

We remark that the bounding functions \(\mathcal {B}_{p}\) that we defined in (19) for the Minkowski case do not satisfy (18).

This lemma can now be used to prove that a primitive of a pole form of the first kind factorizes into a continuous map from [0, *b*) to \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) and the primitive of a pure pole form, whose limiting behaviour at 0 we know from Proposition 1.

### Corollary 1

### Proof

*b*) and, in particular, has a limit in \(O^+\left( \mathbb {R}^{n+2}_1\right) \) at 0. Thus, by (6), \(\varGamma _{p}(\xi \ltimes _{\!p} \psi )\) has a limit in \(\mathbb {P}O\left( \mathbb {R}^{n+2}_1\right) \) at 0. The relation (21) follows from (5) because \(\xi \ltimes _{\!p} \psi \) is a gauge transform of \(\psi \) by the map \(\varGamma ^{p}(\xi )\). The limit (22) follows from

From the factorization (21) and Proposition 1, we conclude that the primitives of spacelike pole forms do not have limits at the singularity, while those of degenerate and Minkowski pole forms of the first kind do have limits. The following corollary deals with these limits for the degenerate case. In Lemma 2 and Proposition 2, we then derive an analogous result for Minkowski pole forms of the first and second kind.

### Corollary 2

*b*) with \(\psi -\xi \) bounded for the pure pole form \(\xi \). Choose \(v_0\in \langle v, w\rangle \cap \mathcal {L}^{n+1}\). Then, there is a continuous map \(k:[0,b)\rightarrow \mathcal {L}^{n+1}\) which satisfies

### Proof

*k*(

*p*) to be a continuous lift of \(\varGamma _p^{0}(\xi \ltimes _{\!p} \omega )\,\langle v_0\rangle \). Due to (22), \(k(0)\in \langle v_0\rangle \). The other relation in (24) follows from (25) and \(\varGamma _p^{q}(\psi )=\varGamma _p^{t}(\psi )\varGamma _t^{q}(\psi )\). \(\square \)

Lemma 1 and Corollary 1 do not hold for pole forms of the second kind. In particular, the gauge transforms (20) are in general not integrable and although the factorization (21) also exists in that case, it seems to be of little use because we do not know whether the first factor, \(\varGamma _{p}(\xi \ltimes _{\!p} \psi )\), has a limit at 0. Thus, we pursue a different strategy. The following lemma and proposition are based on ideas from [17].

### Lemma 2

### Proof

^{3}that \(\varGamma _{p}^{t}(\varPsi )\) satisfies

*t*. Then, evaluating (29) at \({\bar{t}}\) and using (31) yield \(2B<\frac{1}{2} 2B+B\), a contradiction. Hence, indeed (30) holds.

To conclude (27), we note that for all \(p\in (0,\tilde{b})\), the map \(t\mapsto \left| \left( \frac{t}{p}\right) ^\zeta \varGamma _{p}^{t}(\varPsi )\right| \) is continuous, has the value \(|\mathrm{id}|\) at \(t=p\) and, by the above, nowhere on (0, *p*] attains the value \(2B\ge 2|\mathrm{id}|>|\mathrm{id}|\). Thus, necessarily (27) holds. \(\square \)

We can now prove the analogue of Corollary 2 for Minkowski pole forms.

### Proposition 2

*b*) with \(\psi -\xi \) bounded for the pure pole form \(\xi \). Denote by \(v_\pm \) eigenvectors of \(v\wedge w\) with eigenvalues \(\pm \zeta \), \(\zeta >0\). Then, there is a continuous map \(k:[0,b)\rightarrow \mathcal {L}^{n+1}\) which satisfiesand is such that

### Proof

First, we note that (34) follows from (33) and one limit in (33) follows from the other because \((\varGamma _p^{t}(\varPsi ))^*=\varGamma _t^{p}(\varPsi )\) and taking the adjoint is continuous. We prove the first limit in (33).

*p*] converging to 0 and consider the sequence \(\left( \tau \mapsto M(t_i,\tau ,p)\right) _{i\in {\mathbb {N}}}\) of functions from (0,

*p*] to \(\mathbb {R}^{(n+2)^2}\). Due to (9), this sequence of functions converges pointwise. Moreover, due to Lemma 2, the assumed boundedness of \(\varPsi -\varvec{\varXi }\) and (8), the sequence of functions is uniformly bounded by some constant. Thus, the conditions of the dominated convergence theorem are satisfied and we concludewhere we used (9). Again due to Lemma 2 and the assumed boundedness of \(\varPsi -\varvec{\varXi }\), the integrand in

*k*is bounded uniformly in

*p*and so the integral in

*k*exists for all \(p\in (0,\tilde{b})\) and Open image in new window . The first identity in (32) also follows from (4) and (36).

*k*has to take values in the light cone because for all \(p\in (0,\tilde{b})\) we have

## 4 Pole of first order

In this section, we investigate the limiting behaviour at 0 of the Darboux and Calapso transforms of a polarized curve \((\langle c\rangle ,Q)\) on (0, *b*), where \(\langle c\rangle \) has a smooth extension to \((-\,\epsilon ,b)\) for some \(\epsilon >0\) and *Q* has a pole of first order at 0. For convenience, we further assume that \(\langle c\rangle \) and *Q* have smooth extensions to some \((0,b+\tilde{\epsilon })\), \(\tilde{\epsilon }>0\).

*Q*by \(-Q\) has the same effect as replacing \(\lambda \) by \(-\lambda \) and there is no restriction on the range or sign of \(\lambda \). With this assumption, we can choose a smooth lift

*c*of \(\langle c\rangle \) such that \(\mathfrak {Q}(t)=\frac{Q(t)}{\Vert {c}^{\prime }(t)\Vert ^2}=-\frac{1}{t}\) and the 1-form associated with \((\langle c\rangle ,Q)\) reads

*b*) and \(\lambda \omega \) is a degenerate pole form. In particular, it is a pole form of the first kind.

With the help of Corollary 1, we can now show that every Calapso transform of \((\langle c\rangle ,Q)\) converges.

### Theorem 1

Let \((\langle c\rangle ,Q)\) be a polarized curve on (0, *b*) such that \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and *Q* has a pole of first order at 0. Then, for all \(p\in (0,b)\) and \(\lambda \in \mathbb {R}\), the \(\lambda \)-Calapso transform \(\langle c_{\lambda ,p}\rangle \) normalized at *p* has a limit at 0.

### Proof

The Darboux transforms of \((\langle c\rangle ,Q)\) converge to \(\langle c(0)\rangle \).

### Theorem 2

*b*) such that \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and

*Q*has a pole of first order at 0. Then, for all \(\lambda \in \mathbb {R}\backslash \{0\}\), the limit of any \(\lambda \)-Darboux transform \(\langle {\hat{c}}\rangle \) of \((\langle c\rangle ,Q)\) at 0 is \(\langle c(0)\rangle \) (Fig. 1),

### Proof

## 5 Pole of second order

*b*), where \(\langle c\rangle \) is smoothly extendible to \((-\,\epsilon ,b)\) for some \(\epsilon >0\) and

*Q*has a pole of second order at 0. In this case, also \(\lambda \omega \) has a pole of second order at 0 and we cannot apply the results of Sect. 3 directly. Instead, we use a gauge transformation

Again, for convenience, we assume that \(\langle c\rangle \) and *Q* have smooth extensions to some \((0,b+\tilde{\epsilon })\), \(\tilde{\epsilon }>0\).

### 5.1 The singular gauge transformation

We assume that the polarization *Q* is of the form \(Q_t=Q(t){{\mathrm{d}}}t^2=\frac{{{\mathrm{d}}}t^2}{t^2}\). Again, this is no restriction of our results: We can assume *Q*(*t*) to be positive because replacing *Q* by \(-Q\) has the same effect as replacing \(\lambda \) by \(-\lambda \), and we make no restriction on the range of \(\lambda \). Furthermore, the Darboux and Calapso transformations are invariant under reparametrizations of \(\langle c\rangle \). Since *Q* has a pole of second order (see Definition 2), there certainly is a parameter *t* for \(\langle c\rangle \) such that \(Q_t=\frac{{{\mathrm{d}}}t^2}{t^2}\) and \(\langle c\rangle \) remains smoothly extendible to \((-\,\epsilon ,b)\).

For the gauge transformation, we use a product \(g=FR\) of a frame \(F:(0,b)\rightarrow O(R^{n+2}_1)\), smoothly extendible to \((-\,\epsilon ,b)\), and a particularly simple singular factor \(R:(0,b)\rightarrow O(R^{n+2}_1)\).

*F*. To this end, let

*c*be the flat lift of \(\langle c\rangle \), for which \(\Vert {c}^{\prime }\Vert ^2=1\). Then, with the above assumptions, \(\mathfrak {Q}(t)=\frac{1}{t^2}\). Let \(N_1,\ldots ,N_{n-1}\) be parallel, orthonormal unit normal fields of

*c*that satisfy

^{4}Open image in new window for \(i=1,\ldots ,n-1\) and let \({\bar{c}}:(0,b)\rightarrow \mathcal {L}^{n+1}\) be the unique map such that Open image in new window and \(\langle c,{\bar{c}}\rangle =\langle {c}^{\prime },N_1,\ldots ,N_{n-1}\rangle ^\perp \). Now define the frame \(F:(0,b)\rightarrow O\left( \mathbb {R}^{n+2}_1\right) \) to map

*t*to the orthogonal transformation

*F*(

*t*) which maps a

*t*-independent pseudo-orthonormal basis \(\{o,\varvec{\iota }, \mathfrak {t},\mathfrak {n}_{1},\ldots ,\mathfrak {n}_{n-1}\}\) of \(\mathbb {R}^{n+2}_1\) to the

*t*-dependent basis \(\{c(t),{\bar{c}}(t),{c}^{\prime }(t),N_1(t),\ldots ,N_{n-1}(t)\}\). The Maurer–Cartan form of

*F*is then of the form

*R*has Maurer–Cartan form

^{5}on (0,

*b*). Thus, \({g}\!\ltimes \! \lambda \omega \) is a pole form.

The next Lemma, which summarizes the algebraic properties of \(\xi _\lambda \), can be verified by direct computation.

### Lemma 3

In particular, for \(\lambda >0\), the pole form \(\lambda \omega \) is of the first kind. For \(\lambda \le 0\), it is of the second kind.

By Proposition 2, Corollary 1 and Proposition 1, for \(1-2\lambda \ge 0\), the primitives \(\varGamma ^{t}_{p}({g}\!\ltimes \! \lambda \omega )\) converge as \(t\rightarrow 0\) while for \(1-2\lambda <0\) they do not have a limit at 0. We treat these cases separately in the next two sections.

### 5.2 The behaviour at the singularity for \(1-2\lambda \ge 0\)

### Theorem 3

*b*) where \(\langle c\rangle \) is regularly extendible to \((-\epsilon ,b)\) and

*Q*has a pole of second order at 0. Then, for every \(\lambda \in \mathbb {R}\backslash \{0\}\) with \(1-2\lambda \ge 0\), all \(\lambda \)-Darboux transforms \(\langle {\hat{c}}\rangle \) of \(\langle c\rangle \) converge to \(\langle c(0)\rangle \) as 0 is approached,

### Proof

This proves the theorem for \(1-2\lambda >0\). For \(1-2\lambda =0\), the proof works completely analogous with Corollary 2 in place of Proposition 2 and (24) instead of (32). \(\square \)

For the Calapso transforms, even less work is necessary.

### Theorem 4

Let \((\langle c\rangle ,Q)\) be a polarized curve on (0, *b*) where \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and *Q* has a pole of second order at 0. Then, all \(\lambda \)-Calapso transforms with \(1-2\lambda \ge 0\) converge as \(t\rightarrow 0\).

### Proof

### 5.3 The behaviour at the singularity for \(1-2\lambda <0\)

In this case, \(\xi _\lambda \) takes values in an algebra of infinitesimal Euclidean rotations. Thus, by Proposition 1, the primitives of \(\xi _\lambda \) do not have a limit at 0 and hence by Corollary 2 neither do the primitives of \({g}\!\ltimes \! \lambda \omega \) converge. From this, it follows that the \(\lambda \)-Calapso transforms of \(\langle c\rangle \) do not converge.

### Theorem 5

*b*) such that \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and

*Q*has a pole of second order at 0. For \(\lambda \in \mathbb {R}\backslash \{0\}\) and \(p\in (0,b)\), the \(\lambda \)-Calapso transform normalized at

*p*tends

^{6}towards the circular motion

### Proof

*b*), the inner product of the bounded lifts

That \(\langle C_{\lambda ,p}\rangle \) is a parametrization of the circle \({g}(p) \, \varGamma _{p}^{0}(\xi _\lambda \ltimes _{\!p} {g}\!\ltimes \! \lambda \omega ) \, \langle o,\varvec{\iota },\mathfrak {t}\rangle \) follows because \(\varvec{\varXi }_\lambda \) takes values in the subalgebra \(\langle (o-\mathfrak {t})\wedge (\varvec{\iota }-\lambda \mathfrak {t})\rangle \): The one-parameter group \(\varGamma _{p}(\xi _\lambda )\) of Euclidean rotations moves the point \(\langle o\rangle \) on the circle \(\langle o,\varvec{\iota }, \mathfrak {t}\rangle \cap \mathcal {L}^{n+1}\) with speed increasing towards infinity as 0 is approached. \(\square \)

### Lemma 4

For \(\mathcal {V}\) an arbitrary subset of \(\mathbb {R}^m\) and \(q\in \mathbb {R}^m\) a limit point of \(\mathcal {V}\), let \(\langle A\rangle :\mathcal {V}\rightarrow \mathbb {P}O(\mathbb {R}^{n+2}_1)\) be a map with limit \(\langle xy^*\rangle \) at *q*, where \(x,y\in \mathcal {L}^{n+1}\). Let \(\langle u\rangle :\mathcal {V}\rightarrow S^n\) be a map so that there is an open neighbourhood *U* of \(\langle y\rangle \in S^n\) such that \(\langle u(\mathcal {V})\rangle \cap U\) is empty. Then the product \(\langle A\rangle \langle u\rangle \) has the limit \(\langle x\rangle \) at *q* (Fig. 2).

### Proof

Let *A* be a lift of \(\langle A\rangle \) with limit \(xy^*\) at *q* and *u* the lift of \(\langle u\rangle \) that satisfies Open image in new window for all \(t\in \mathcal {V}\). Then, *u* is bounded with respect to the positive-definite norm \(|{\cdot }|\) by some \(R\in \mathbb {R}^+\). Namely, if there was no such bound, then in order to maintain Open image in new window , the point \(\langle u(t)\rangle \) would need to get arbitrarily close to \(\langle y\rangle \), which contradicts the assumptions.

*u*is bounded by \(R\in \mathbb {R}\), we have

*q*. Thus, the product

*Au*has the limit \(-x\) at

*q*and \(\langle A\rangle \langle u\rangle \) has the limit \(\langle x\rangle \) at

*q*. \(\square \)

### Proposition 3

Let \((\langle c\rangle ,Q)\) be a polarized curve on (0, *b*) such that \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and *Q* has a pole of second order at 0. For \(\lambda \in \mathbb {R}\) with \(1-2\lambda < 0\) and \(\langle {\hat{c}}_p\rangle \) not on the limit circle \(\pounds [\langle c_{\lambda ,p}\rangle ]\) of the \(\lambda \)-Calapso transform normalized at *p* (see (44)), denote by \(\langle {\hat{c}}\rangle \) the \(\lambda \)-Darboux transform of \(\langle c\rangle \) with initial value \(\langle {\hat{c}}(p)\rangle =\langle {\hat{c}}_p\rangle \). Then, \(\langle {\hat{c}}\rangle \) converges to \(\langle c(0)\rangle \) as 0 is approached.

### Proof

*V*of \(\langle W\rangle \) and

*U*of \(\langle \varvec{\iota }\rangle \) such that

The characterization of the \(\lambda \)-Darboux transforms excluded in Proposition 3 is independent of the point \(p\in (0,b)\), at which the initial condition \(\langle {\hat{c}}(p)\rangle =\langle {\hat{c}}_p\rangle \) is posed. As soon as \(\langle {\hat{c}}(p)\rangle \in \pounds [\langle c_{\lambda ,p}\rangle ]\) holds for one \(p\in (0,b)\), it holds for all \(p\in (0,b)\).

We finally analyse what happens, when the initial point of the Darboux transform does lie on the limit circle of the Calapso transform. In this case, we cannot apply Lemma 4 and indeed \(\langle {\hat{c}}\rangle \) does not converge to \(\langle c(0)\rangle \) as 0 is approached.

### Theorem 6

*b*) where \(\langle c\rangle \) has a regular extension to \((-\,\epsilon ,b)\) and

*Q*has a pole of second order at 0. As 0 is approached, any \(\lambda \)-Darboux transform \(\langle {\hat{c}}\rangle \) with \(1-2\lambda <0\) converges to \(\langle c(0)\rangle \),

### Proof

The first part of the theorem was proved above (see Proposition 3). What remains to be confirmed is the exception. The analysis of this exceptional case is laborious and will occupy us for the rest of this section. We prove the theorem in three steps. First, we show that \(\langle {\hat{c}}\rangle \) approaches the curvature circle of \(\langle c\rangle \) at 0. Then, we show that \(\langle {\hat{c}}\rangle \) has at least two distinct limit points on that curvature circle at 0. In the third step, we use that \(\langle {\hat{c}}\rangle \) and \(\langle c\rangle \) envelop a circle congruence to argue that \(\langle {\hat{c}}\rangle \) tends towards a rotation on the curvature circle of \(\langle c\rangle \) at 0 with frequency increasing towards infinity as 0 is approached.

*b*).

- 1.Denote by \(\varvec{\varXi }_\lambda \) the orthogonal lift of \(\xi _\lambda \) and let \({\hat{c}}\) be the liftfor some nonzero \({\hat{C}}_p\in \langle {\hat{C}}_p\rangle \). In Sect. 5.1, we chose the normal fields \(\{N_1,\ldots ,N_{n-1}\}\) such that \(\langle N_1,\ldots ,N_{n-1}\rangle ^\perp \) is the curvature circle congruence of \(\langle c\rangle \). Therefore, since \({\hat{c}}\) is a lift of \(\langle {\hat{c}}\rangle \) that stays away from zero, in order to show that \(\langle {\hat{c}}\rangle \) approaches the curvature circle of \(\langle c\rangle \) at 0, it is sufficient to show that the \(n-1\) inner products Open image in new window all tend towards zero. Since \({g}^{-1}N_i=\mathfrak {n}_i\) and \(\varvec{\varXi }_\lambda (\mathfrak {n}_i)=0\) (see Sect. 5.1), the inner product of (53) with \(N_i\) reads Since \({\hat{C}}_p\) is an element of \(\langle o,\varvec{\iota },\mathfrak {t}\rangle \) and \(\mathfrak {n}_i\in \langle o,\varvec{\iota },\mathfrak {t}\rangle ^\perp \), all the functions \(\sigma _i(t)\) tend to zero as \(t\rightarrow 0\). Since \(\varPhi (t)/t\) is bounded, \(\varPhi (t)\) tends to zero as 0 is approached and thus also the first derivatives of all \(\sigma _i\) tend to zero as 0 is approached. An application of the mean value theorem to \(\sigma _i\) then yields that Open image in new window tends to zero and indeed \(\langle {\hat{c}}\rangle \) approaches the curvature circle of \(\langle c\rangle \) at 0 as 0 is approached.$$\begin{aligned} {\hat{c}}(t)=\frac{1}{t}{g}(t) \, \varGamma _t^{p}(\varvec{\varXi }_\lambda ) \, \varGamma _t^{0}(\varPhi ) \, {\hat{C}}_p \end{aligned}$$(53)
- 2.
We now prove that \(\langle {\hat{c}}\rangle \) has at least two limit points at 0, namely \(\langle c(0)\rangle \) and \(\langle {\bar{c}}(0)\rangle \), where \({\bar{c}}\) is as in Sect. 5.1.

To prove that \(\langle c(0)\rangle \) is a limit point, let \((t_i)_{i\in {\mathbb {N}}}\) be a sequence in (0,*b*) which converges to 0 and is such thatIn particular,$$\begin{aligned} \forall i\in {\mathbb {N}}:~\varGamma _{t_i}^{p}(\xi _\lambda ) \, \langle {\hat{C}}_p\rangle = \langle o\rangle . \end{aligned}$$Then, evaluating (52) at \(t_i\) and taking the limit \(i\rightarrow \infty \) yield$$\begin{aligned} \forall i,j>0:~\varGamma _{t_i}^{t_j}(\xi _\lambda )=\mathrm{id}. \end{aligned}$$Thus, \(\langle c(0)\rangle \) is a limit point of \(\langle {\hat{c}}\rangle \).$$\begin{aligned}&\lim _{i\rightarrow \infty }\langle {\hat{c}}(t_i)\rangle =\lim _{i\rightarrow \infty }\langle {g}(t_i)\rangle \varGamma _{t_i}^{p}(\xi _\lambda )\varGamma _{t_i}^{0}(\phi )\langle {\hat{C}}_p\rangle \\&\quad =\langle c(0)\varvec{\iota }^*\rangle \varGamma _{t_1}^{0}(\xi _\lambda )\langle {\hat{C}}_p\rangle =\langle c(0)\varvec{\iota }^*\rangle \langle o\rangle =\langle c(0)\rangle . \end{aligned}$$To prove that also \(\langle {\bar{c}}(0)\rangle \) is a limit point, rescale \({\hat{C}}_p\) if necessary and let \((t_i)_{i{\mathbb {N}}}\) be a sequence in (0,*b*) which converges to 0 and is such thatThe lift (53) evaluated at \(t_i\) yields the sequence \(({\hat{c}}(t_i))_{i\in {\mathbb {N}}}\) with$$\begin{aligned} \forall i\in {\mathbb {N}}:~\varGamma _{t_i}^{p}(\varvec{\varXi }_\lambda ) {\hat{C}}_p= \varvec{\iota }. \end{aligned}$$(54)We have already proved above that the \(n-1\) quantities Open image in new window converge to zero. We now show that also both Open image in new window and Open image in new window converge to zero and that Open image in new window converges to \(-1\). It then follows that \(\langle {\hat{c}}(t_i)\rangle \) converges to \(\langle {\bar{c}}(0)\rangle \).$$\begin{aligned} {\hat{c}}(t_i)=\frac{1}{t_i}{g}(t_i) \, \varGamma _{t_1}^{p}(\varvec{\varXi }_\lambda ) \, \varGamma _{t_i}^{0}(\varPhi ) \, {\hat{c}}_p. \end{aligned}$$Use \(g^{-1}(t){\bar{c}}(t)=\frac{1}{t} \varvec{\iota }\) and (54) to write Now use the integral Eq. (7) for \(\varGamma _{t}^{p}(\varPhi )\) twice and the antisymmetry of elements of \(\mathfrak {o}\left( \mathbb {R}^{n+2}_1\right) \) to get Since \(\varPhi (t)/t\) is bounded, indeed Open image in new window converges to zero as \(i\rightarrow \infty \). To show that the inner product of \({\hat{c}}(t_i)\) with \({c}^{\prime }(t_i)/\Vert {c}^{\prime }(t_i)\Vert \) converges to zero, we write Again use the integral equation for \(\varGamma ^{0}(\varPhi )\), the boundedness of \(\varPhi /t\) and (54) to conclude that Open image in new window converges to zero as \(i\rightarrow \infty \).Finally, which converges to Open image in new window .Thus, indeed \(\lim _{i\rightarrow \infty }{\hat{c}}(t_i)={\bar{c}}(0)\) such that \(\langle {\bar{c}}(0)\rangle \) is also a limit point of \(\langle {\hat{c}}\rangle \).

- 3.
We now show that \(\langle {\hat{c}}\rangle \) tends towards a rotation on the curvature circle of \(\langle c\rangle \) at 0. Since both \(\langle c(0)\rangle \) and \(\langle {\bar{c}}(0)\rangle \) are limit points of \(\langle {\hat{c}}\rangle \), it then follows that the frequency of this rotation must tend towards infinity.

Choose \(\tilde{b}\in (0,b)\) such that \(\langle {\bar{c}}(0)\rangle \notin \langle c((0,\tilde{b}))\rangle \). Then, there is a closed neighbourhood \(U\subset S^n\) of \(\langle {\bar{c}}(0)\rangle \) whose intersection with \(\langle c((0,\tilde{b}))\rangle \) is empty. DefineSince \(\langle {\bar{c}}(0)\rangle \) is a limit point of \(\langle {\hat{c}}\rangle \), the set \(\mathcal {I}\) is non-empty and has the limit point 0. Now let \({\bar{c}}\),$$\begin{aligned} \mathcal {I}:=\{t\in (0,\tilde{b})~|~\langle {\hat{c}}(t)\rangle \in U\}. \end{aligned}$$*c*and \({\hat{c}}\) be spherical lifts of \(\langle {\bar{c}}\rangle \), \(\langle c\rangle \) and \(\langle {\hat{c}}\rangle \), respectively. The circle congruence \(\mathcal {S}\) enveloped by \(\langle {\hat{c}}\rangle \) and \(\langle c\rangle \) is We have showed above that \(\varPi _{\langle N_1,\ldots ,N_{n-1}\rangle }{\hat{c}}\) converges to zero as 0 is approached. Moreover, we have constructed \(\mathcal {I}\) in a way that there exists a constant \(C>0\) such that Therefore, \(\mathcal {S}\) restricted to \(\mathcal {I}\) converges to the curvature circle \(\langle {\bar{c}}(0),c(0),{c}^{\prime }(0)\rangle \) of \(\langle c\rangle \) at 0. (55) also implies that inside*U*the speed of \(\langle {\hat{c}}\rangle \) tends towards infinity because Now stereographically project \(S^n\) to \(\mathbb {R}^n\) such that the curvature circle of \(\langle c\rangle \) at 0 gets mapped to the unit circle in the \(\langle e_1,e_2\rangle \)-plane with centre the origin. Write the projection \(\hat{\mathfrak {c}}\) of \(\langle {\hat{c}}\rangle \) aswhere \(\varphi :(0,b)\rightarrow \mathbb {R}\) is smooth and \(Z(t)\in \langle e_3,\ldots ,e_n\rangle \) for all \(t\in (0,b)\). Since \(\langle {\hat{c}}\rangle \) approaches the curvature circle of \(\langle c\rangle \) at 0, the function$$\begin{aligned} \hat{\mathfrak {c}}(t)=\rho (t)\left( \sin (\varphi (t))e_1+\cos (\varphi (t))e_2\right) +Z(t), \end{aligned}$$*Z*must converge to zero and \(\rho \) must converge to 1 as \(t\rightarrow 0\). Since moreover the enveloped circle congruence \(\mathcal {S}\) restricted to \(\mathcal {I}\) converges to the curvature circle of \(\langle c\rangle \) at 0, on \(\mathcal {I}\) the angle between the tangent to \(\hat{\mathfrak {c}}\) and the tangent to the circle \(\sin (\varphi )e_1+\cos (\varphi )e_2\) must converge to \(\pm 1\), that is, Due to the lower bound (56), the square root in this expression tends to infinity and so also \({\varphi }^{\prime }\) must converge to plus or minus infinity as 0 is approached in \(\mathcal {I}\). Together with the result that also \(\langle c(0)\rangle \) is a limit point of \(\langle {\hat{c}}\rangle \), this implies that \(\varphi \) converges to plus or minus infinity such that indeed the frequency of the rotation of \(\langle {\hat{c}}\rangle \) on the curvature circle tends towards infinity (Fig. 3).

## Footnotes

- 1.
One can easily show that \(\langle {\hat{c}}_p\rangle \) does not lie on the image of \(\langle c_{\lambda ,p}\rangle \) if and only if \(\langle {\hat{c}}(t)\rangle \) is different from \(\langle c(t)\rangle \) for all \(t\in (a,b)\), which is the analogue of condition (1) of Definition 1.7 in [8] for polarized curves.

- 2.
Our parameter \(\lambda \) is \(-\,2\) times the parameter

*t*used in [6]. - 3.
Or by using the integral Eq. (7) for the 1-form \(\xi \ltimes _{\!p} \varPsi \) and the identity \(\varGamma _p(\xi \ltimes _{\!p} \varPsi )=\varGamma _p(\varPsi )\varGamma ^{p}(\varvec{\varXi })\).

- 4.
This condition means that \(\langle N_1,\ldots ,N_{n-1}\rangle ^\perp \cap \mathcal {L}^{n+1}\) is the curvature circle congruence of \(\langle c\rangle \).

- 5.
In fact, even \(\frac{1}{t} |({g}\!\ltimes \! \lambda \omega -\xi _\lambda )_t|\) is bounded on (0,

*b*), which will be useful in the proof of Theorem 6. - 6.
\(\langle c_{\lambda ,p}\rangle \) tends towards \(\langle C_{\lambda ,p}\rangle \) in the sense that the distance between \(\langle c_{\lambda ,p}(t)\rangle \) and \(\langle C_{\lambda ,p}(t)\rangle \) measured with respect to an arbitrary representative metric of the conformal structure of \(S^n\) tends to zero as \(t\rightarrow 0\).

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The author would like to thank G. Szewieczek for fruitful discussions and S. Fujimori for his hospitality during a visit to Okayama University, where parts of this work were elaborated. Special thanks go to U. Hertrich-Jeromin for his numerous, valuable comments and his support as supervisor of the author’s Ph.D. Thesis. This work has been supported by the Austrian Science Fund (FWF) and the Japan Society for the Promotion of Science (JSPS) through the FWF/JSPS Joint Project Grant I1671-N26 “Transformations and Singularities.”

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