Advances in Discrete Differential Geometry pp 309-345 | Cite as

# Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization

## Abstract

In this article, we study an analog of the Björling problem for isothermic surfaces (that are a generalization of minimal surfaces): given a regular curve \(\gamma \) in \(\mathbb {R}^3\) and a unit normal vector field *n* along \(\gamma \), find an isothermic surface that contains \(\gamma \), is normal to *n* there, and is such that the tangent vector \(\gamma '\) bisects the principal directions of curvature. First, we prove that this problem is uniquely solvable locally around each point of \(\gamma \), provided that \(\gamma \) and *n* are real analytic. The main result is that the solution can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is read off from \(\gamma \), and then passing to the limit of vanishing mesh size. The proof relies on a rephrasing of the Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its discretization which is induced from the geometry of discrete isothermic surfaces. The discrete-to-continuous limit is carried out for the Christoffel and the Darboux transformations as well.

## 1 Introduction

Isothermic surfaces are among the most classical objects in differential geometry: these are surfaces that admit a conformal parametrization along curvature lines, see Definition 1. Like various particular geometries—special coordinate systems, minimal surfaces, surfaces of constant curvature—they have been introduced and intensively studied in the second half of the 19th century [9, 24]. Also, like the many of these classical objects, they have been “rediscovered” in the 1990s, both in connection with integrable systems and in the context of discrete differential geometry. The first description of isothermic surfaces as soliton surfaces is found in [11]. The first definition for discrete isothermic surfaces was made shortly after in [3]. In the most simple case, these are immersions of \({\mathbb {Z}}^2\) into \({\mathbb {R}}^3\) such that the vertices of each elementary quadrilateral are conformally equivalent to the corners of a planar square, see Definition 3.

This discrete surface class, its transformations and invariances has been studied e.g. in [7, 8, 20]. A systematic presentation of the theory of isothermic surfaces in the context of Möbius geometry can be found in [15]. Finally, we refer to [4] for a detailed overview on discrete isothermic surfaces as part of discrete differential geometry, including historical remarks.

^{1}The appearance of an elliptic equation suggests that data for the surface boundary should be prescribed, as it is done for minimal surfaces for example. The hyperbolic equations, on the other hand, suggest to provide data for two curvature lines instead, like in the case of level surfaces in triply orthogonal systems [1]. Neither of the two approaches seems promising for the coupled system.

In contrast, there is a canonical way to pose an initial value problem for a *discrete* isothermic surface. One prescribes the vertices in \(\mathbb {R}^3\) for a “zig-zag”-curve in parameter space as indicated in Fig. 1. For vertices in general position, these data can be extended to a discrete isothermic surface in a unique way. In fact, all vertices on the discrete surface are easily obtained inductively from the prescribed data.

*In this paper, we formulate and prove solvability of a* Björling problem *for real analytic isothermic surfaces. And we prove that the solution can be obtained as the continuous limit of discrete isothermic surfaces.*

The classical Björling problem is to find a minimal surface that touches a given curve in \(\mathbb {R}^3\) along prescribed tangent planes. This problem has been solved in general, see [13]. An extension of the Björling problem to surfaces of constant mean curvature has been posed (and solved) in [5]. A natural formulation of the Björling problem in the yet more general class of isothermic surfaces reads as follows.

### Problem 1

Given a regular curve \(\gamma \) in \(\mathbb {R}^3\), and two mutually orthogonal unit vector fields *v*, *w* along \(\gamma \), neither of which is tangent to \(\gamma \) at any point. Find an isothermic surface *S* containing \(\gamma \) such that *v* and *w* are the principal directions of curvature at each point of \(\gamma \).

We believe that this problem is solvable locally provided that \(\gamma \), *v* and *w* are all real analytic. Indeed, the non-tangency of the vector fields allows to give a reformulation in terms of a non-characteristic Cauchy problem. However, we do not address the Björling problem in this full generality here, but stick to the following restricted setting, where we do not prescribe two tangent vector fields *v* and *w* individually, but only a two-dimensional tangent plane:

### Problem 2

Given a regular curve \(\gamma \) in \(\mathbb {R}^3\), and a unit vector field *n* that is orthogonal to the tangent vectors \(\gamma '\) at each point. Find an isothermic surface *S* containing \(\gamma \) such that at each point of \(\gamma \), the vector *n* is normal to *S*, and each of the two directions of principal curvature encloses an angle \(\pi /4\) with \(\gamma '\).

As a corollary of the results presented here, it follows that this problem is uniquely solvable for real analytic \(\gamma \) and *n*, at least locally around each point of \(\gamma \). Existence and uniqueness of a real analytic isothermic surface *S* for given data is the minor result of this paper, see Theorem 1. The main result is that the real analytic data can be “sampled” with a mesh width \(\epsilon >0\) in a suitable way such that the discrete isothermic surfaces \(S^\epsilon \) constructed from the discrete data converge in \(C^1\) to *S*. The precise formulation is given in Theorem 2.

It is remarkable that naive numerical experiments suggest that such an approximation result might *not* be true. It was already noted in [3] that discrete isothermic surfaces depend very sensitively on their initial data. The limit \(\epsilon \rightarrow 0\) is delicate, and inappropriate choices of the initial zig-zag cause the sequence \(S^\epsilon \) to diverge rapidly. In fact, even the possibility to construct *any* sequence of discrete isothermic surfaces that approximates a given smooth one is not obvious. Discrete isothermic surfaces are one of many examples of a discretized geometric structure for which the passage back to the original continuous structure needs a highly non-trivial approximation result, the proof of which is analysis-based and goes far beyond elementary geometric considerations. Further such non-trivial convergence results are available, for instance, for discrete surfaces of constant negative Gaussian curvature [2], for discrete triply orthogonal systems [1], and, most importantly, for circle patterns [6, 21, 22] as approximations to conformal maps.

The core of our convergence proof is a stability analysis of the *discrete Gauss-Codazzi system* that we derive for discrete isothermic surfaces. We show that the solution to the discrete Gauss-Codazzi equations with sampled data as initial condition remains close to the solution of the classical Gauss-Codazzi system for the same (continuous) initial data. In a second step, this implies proximity of the respective discrete and continuous surfaces. We are able to quantify the approximation error in terms of the supremum-distance between analytic functions on complex domains: it is linear in the mesh size. In fact, we conjecture that this result is sub-optimal, and second-order approximation should be provable, using a more refined analysis and a more careful approximation of the data.

The techniques used in the proof are similar to those employed by one of the authors [17] to prove convergence of circle patterns to conformal maps. The geometric situation for isothermic surfaces, however, is much more complicated, and the structure of the Gauss-Codazzi system is much more complex than the Cauchy-Riemann equations. The proof of stability relies on estimates for the solution of analytic Cauchy problems in scales of Banach spaces. These estimates have been developed—in the classical, non-discretized setting—in Nagumo’s famous article [18] as part of the existence proof for analytic Cauchy problems. Here, we shall rather use Nirenberg’s [19] version of these estimates. For an overview over the history of analytic Cauchy problems and the related estimates, see the beautiful article of Walter [23].

Note that the convergence proof here is more direct than the one in [17]. While the latter was based on purely discrete considerations, the current proof uses semi-discrete techniques: a–somewhat artificial–extension of the discrete functions to continuous domains allows to formulate estimates more easily. The main simplification, however, is that we separate the proofs for existence of a classical solution and its approximation by discrete solutions.

The paper is organized as follows. In Sect. 2 we formulate the Gauss-Codazzi-system for smooth isothermic surfaces in the framework of analytic Cauchy problems and prove unique local solvability of the Björling problem by the Cauchy-Kowalevskaya theorem. In Sect. 3 we derive an analogous system of difference equations for discrete isothermic surfaces. For appropriate initial conditions, the convergence of the discrete solutions to the corresponding smooth ones is proven in Sect. 4. Then, in Sect. 5 we explain how to discretize the Björling initial data appropriately, and prove convergence of the discrete surfaces to the respective smooth one. Finally, in Sect. 6, the convergence result is extended to Christoffel and Darboux transformations.

## 2 Smooth Isothermic Surfaces

We start by summarizing basic properties of smooth isothermic surfaces and proving our first result on the local solvability of the Björling problem.

### 2.1 Coordinates and Domains

*x*,

*y*) on \(\mathbb {R}^2\) simultaneously. These coordinates are related to each other by

*x*,

*y*) as the auxiliary ones. More precisely: in the rare cases that we need to specify explicitly the arguments of a function \(g:\varOmega \rightarrow \mathbb {R}\) defined on a domain \(\varOmega \subset \mathbb {R}^2\), then we shall write \(g(\xi ,\eta )\) for the value of

*g*at the point with coordinates \(x=\eta +\xi \) and \(y=\eta -\xi \).

*x*,

*y*)-coordinates, \(\varOmega (r|h)\) is a axes-parallel square of side length 2

*r*, centered at the origin, that is cut off at the top-right and bottom-left corners.

### 2.2 Definition and Equations

### Definition 1

*isothermic surface*, if

- (1)
*F*is conformal, i.e., there exists a conformal factor \(u:\varOmega (r|h)\rightarrow \mathbb {R}\) such that$$\begin{aligned} \Vert F_x\Vert ^2 = \Vert F_y\Vert ^2 = e^{2u},\quad \langle F_x,F_y \rangle = 0, \end{aligned}$$(3) - (2)
*F*parametrizes along curvature lines, i.e., the normal map \(N:\varOmega (r|h)\rightarrow \mathbb {S}^2\) satisfies$$\begin{aligned} \langle F_{xy},N\rangle = 0. \end{aligned}$$(4)

### Remark 1

The genuine principal curvature functions are given by \(e^{-u}\mathfrak {k}\) and \(e^{-u}\mathfrak {l}\). The quantities \(\mathfrak {k}\) and \(\mathfrak {l}\) are better suited for the calculations below.

The next result is classical.

### Lemma 1

*F*is uniquely determined up to Euclidean motions.

We briefly recall the proof, since we shall need some of the calculations later.

### Proof

*U*and

*V*:

*U*and

*V*imply that

*F*is non-degenerate, and it is uniquely determined by its value \(F(0)\in \mathbb {R}^3\) at \((x,y)=0\). Clearly \(\varPsi \) is an adapted frame for the surface defined by

*F*, whose normal vector field is given by \(\varPsi _3\). It follows directly from (8) that

*F*is conformal (3). The property (4) is a further direct consequence of (6) and the special form of

*U*and

*V*from (7). \(\square \)

*x*,

*y*) and \((\xi ,\eta )\) by \(x=\eta +\xi \) and \(y=\eta -\xi \) in Sect. 2.1. Introduce auxiliary functions \(v,w:\varOmega (r|h)\rightarrow \mathbb {R}\) by

### 2.3 Local Solution of the Björling Problem

The following result implies local solvability of (the restricted version of) the Björling problem for isothermic surfaces, with real analytic data. To see the equivalence to Problem 2 stated in the introduction, observe that the conformal parametrization *F* of an isothermic surface and our coordinates in (1) are such that the images of \(\{x=\mathrm {const}\}\) and of \(\{y=\mathrm {const}\}\) are mapped to curvature lines under *F*, whereas the tangent to each curve \(\xi \mapsto F(\xi ,\eta )\) is always at an angle of \(\pi /4\) to both curvature directions.

### Theorem 1

*F*and its normal vector field

*N*satisfy

### Remark 2

The original Björling problem consists in finding a minimal surface in \(\mathbb {R}^3\) that touches a given curve along prescribed tangent planes. See [5] for an extension to constant mean curvature surfaces. Our problem is a bit different since (13) implies in addition that the tangential vector to the data curve is everywhere at angle \(\pi /4\) with the directions of principal curvature, see (14). Such additional restrictions are expected to guarantee unique solvability of the Björling problem in the much larger class of isothermic surfaces.

### Proof

*f*and

*n*determine both the conformal factor \(u\) and the adapted frame \(\varPsi =(e^{-u}F_x,e^{-u}F_y,N):\varOmega (r|h) \rightarrow \mathrm {SO(3)}\) uniquely on \(\eta =0\); denote the corresponding functions by \(u^0:(-r,r)\rightarrow \mathbb {R}\) and \(\varPsi ^0:(-r,r)\rightarrow \mathrm {SO(3)}\), respectively.

*F*is clear from its construction in the proof. To see that

*F*attains the initial data (13), first observe that an adapted frame \(\varPsi \) necessarily satisfies \(\varPsi _\xi =\varPsi _x-\varPsi _y=\varPsi (U-V)\), and so \(\varPsi =\varPsi ^0\) on \(\eta =0\), thanks to (7) and (15), (16). In particular, we have that \(\varPsi _3(\xi ,0)=N(\xi )\). And further, \(F_\xi = \varPsi _1-\varPsi _2 =\varPsi _1^0-\varPsi _2^0\) implies \(F=f\) on \(\eta =0\).

Concerning uniqueness: *f* and \(\varPsi ^0\) determine the initial data \((v^0,w^0,\mathfrak {k}^0,\mathfrak {l}^0)\) for (9)–(12)—and hence also its solution \((v,w,\mathfrak {k},\mathfrak {l})\)—uniquely. Invoking again Lemma 1, it follows that *F* with the normalization (16) is unique as well . \(\square \)

## 3 Discrete Isothermic Surfaces

*z*, assuming that \(|\epsilon z|<1\).

### 3.1 Coordinates and Domains

*x*,

*y*) being the “auxiliary” ones. Introduce the associated shift-operators \({{T}}_x,{{T}}_y,{{T}}_\xi ,{{T}}_\eta \) by

*f*is defined on \(\varOmega (r|h)\). Likewise, we define \(\varOmega ^{[y]\epsilon }(r|h)\). The domain

*elementary*\(\epsilon \)-

*square*.

### 3.2 Definition of Discrete Isothermic Surfaces

In this section, we give a variant of the definition for discrete isothermic surfaces from [3], which is well-suited for the passage to the continuum limit. First, we need auxiliary notation.

### Definition 2

*(non-degenerate) conformal*

*square*iff they lie on a circle, but no three of them are on a line, they are cyclically ordered,

^{2}and their mutual distances are related by

### Remark 3

Alternatively, one could define conformal squares by saying that \(p_1\) to \(p_4\) have cross-ratio equal to minus one, either in the sense of quaternions, see for example [15], or after identification of these points with complex numbers in their common plane. Again, non-degeneracy is important for equivalence of the definitions.

The following is an easy exercise in elementary geometry.

### Lemma 2

For any given three points \(p_1,p_2,p_3\in \mathbb {R}^3\) (with ordering) that are not collinear (and in particular pairwise distinct), there exists precisely one fourth point \(p_4\in \mathbb {R}^3\) that completes the conformal square. Moreover, the coordinates of \(p_4\) depend analytically on those of \(p_1\), \(p_2\) and \(p_3\).

We are now going to state the main definition, namely the one for discrete isothermic surfaces. Originally [3], discrete surfaces have been introduced as particular immersed lattices in \(\mathbb {R}^3\). Having the continuous limit in mind, we give a slightly different definition, which describes a continuous immersion in \(\mathbb {R}^3\), corresponding to a two-parameter family of lattices.

### Definition 3

A map \(F^\epsilon :\varOmega (r|h)\rightarrow \mathbb {R}^3\) is called (the parametrization of) a \(\epsilon \)-*discrete isothermic surface*, if elementary \(\epsilon \)-squares are mapped to conformal squares in \(\mathbb {R}^3\).

### Remark 4

### 3.3 The Discrete Björling Problem

We introduce the analog of the Björling problem for \(\epsilon \)-discrete isothermic surfaces. In contrast to its continuous counterpart, its solution is immediate. First, we need some more notation to formulate conditions on the data.

### Definition 4

*non-degenerate*if neither any of the point triples

*degenerate*.

### Definition 5

We call a function \(f^\epsilon :\varOmega (r|\frac{\epsilon }{2})\rightarrow \mathbb {R}^3\) *Björling data* for the construction of an \(\epsilon \)-discrete isothermic surface if it is non-degenerate.

### Proposition 1

Let \(\bar{h}\) and \(\epsilon >0\) with \(r>\bar{h}>\frac{\epsilon }{2}\) and Björling data \(f^\epsilon \) be given. Then, there exists some maximal \(h\in (\frac{\epsilon }{2},\bar{h}]\) and a unique \(\epsilon \)-discrete isothermic surface \(F^\epsilon :\varOmega (r|h)\rightarrow \mathbb {R}^3\) such that \(F^\epsilon =f^\epsilon \) on \(\varOmega (r|\frac{\epsilon }{2})\). Here *maximal* has to be understood as follows: either \(h=\bar{h}\), or the restriction of \(F^\epsilon \) to \(\varOmega (r|h-\frac{\epsilon }{2})\) is degenerate.

### Proof

The proof is a direct application of Lemma 2: from the data \(f^\epsilon \) given on \(\varOmega (r|\frac{\epsilon }{2})\), one directly calculates the values of \(F^\epsilon \) on \(\varOmega (r|\epsilon )\). These are then extended to \(\varOmega (r|3\frac{\epsilon }{2})\) in the next step, and so on. The procedure works as long as no degeneracies occur. \(\square \)

### 3.4 Discrete Quantities and Basic Relations

Let some discrete isothermic surface \(F^\epsilon :\varOmega (r|h)\rightarrow \mathbb {R}^3\) be given. Below, we introduce quantities that play an analogous role for \(F^\epsilon \) as \(u\), \(\mathfrak {k}\), \(\mathfrak {l}\) etc. do for *F*. Figure 3 *(right)* indicates, on which lattices these respective quantities live.

*discrete conformal factors*\({\hat{u}}:\varOmega ^{[x]\epsilon }(r|h)\rightarrow \mathbb {R}\) and \({\check{u}}:\varOmega ^{[y]\epsilon }(r|h)\rightarrow \mathbb {R}\), respectively, by

*discrete derivatives*\(v,w:\varOmega ^{[xy]\epsilon }(r|h)\rightarrow \mathbb {R}\) of the conformal factor by

*discrete unit tangent vectors*\(a:\varOmega ^{[x]\epsilon }(r|h)\rightarrow \mathbb {S}^2\) and \(b:\varOmega ^{[y]\epsilon }(r|h)\rightarrow \mathbb {S}^2\), respectively, by

*normal field*\(N:\varOmega ^{[xy]\epsilon }(r|h)\rightarrow \mathbb {S}^2\), namely

*discrete scaled principal curvatures*\(\mathfrak {k}:\varOmega ^{[xxy]\epsilon }(r|h)\rightarrow \mathbb {R}\) and \(\mathfrak {l}:\varOmega ^{[xyy]\epsilon }(r|h)\rightarrow \mathbb {R}\), respectively, by

*v*,

*w*) and \(({\tilde{v}},{\tilde{w}})\) are just different representations of the same geometric information.

### Lemma 3

*v*,

*w*) and \(({\tilde{v}},{\tilde{w}})\) of functions. Specifically, recalling the \({}^*\)-notation introduced in (17),

### Proof

*w*and

*v*exchanged. Clearly, these equations are uniquely solvable for \(({\tilde{v}},{\tilde{w}})\) in terms of (

*v*,

*w*):

*v*, \({\tilde{v}}\) have the same sign, and

*w*, \({\tilde{w}}\) have the same sign by (25).

Finally, to calculate \({\tilde{v}}\) from a given \((v,{\tilde{w}})\) using the first relation in (23), it suffices to invert the (strictly increasing) function \({\tilde{v}}\mapsto {\tilde{v}}/{\tilde{v}}^*\). Then, knowing \({\tilde{v}}\) and \({\tilde{w}}\), the value of *w* can be obtained from the second relation in (23). \(\square \)

Recall that all discrete quantities defined above depend on the parameter \(\epsilon \). To stress this fact, we will in the following use the superscript \(\epsilon \).

For later reference, we draw some first consequences of the definitions above. Specifically, we summarize the relations between the geometric quantities \((a^\epsilon ,b^\epsilon ,{\hat{u}}^\epsilon ,{\check{u}}^\epsilon )\), and, of course, to \(F^\epsilon \) itself, to the more abstract quantities \((v^\epsilon ,w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon )\) that satisfy the Gauss-Codazzi system (31)–(34). These relations can be seen as a discrete analog of the frame equations (6) and (7).

### Lemma 4

### Proof

### 3.5 Discrete Gauss-Codazzi System

This section is devoted to derive a discrete version of the Gauss-Codazzi equations (9)–(12). The following definition is needed to classify the difference between the continuous and the discrete system.

### Definition 6

*asymptotically analytic on*\(\mathbb {C}^n\) if the following is true. For every \(M>0\), there is an \(\epsilon (M)>0\) such that each \(h_\epsilon \) with \(0<\epsilon <\epsilon (M)\) extends from \(D_\epsilon \) to a complex-analytic function \(\tilde{h}_\epsilon :{\mathbb D}_M^n\rightarrow \mathbb {C}\) on the

*n*-dimensional complex multi-disc

The prototypical example for a family \((h_\epsilon )_{\epsilon >0}\) that is asymptotically analytic on \(\mathbb {C}\) is given by \(h_\epsilon (z)=1/z^*=(1-\epsilon ^2z^2)^{-1/2}\). It is further easily seen that also the functions \(g_\epsilon =\epsilon ^{-2}(h_\epsilon -1)\) form such a family; this is a very strong way of saying that \(h_\epsilon =1+\mathscr {O}(\epsilon ^2)\).

### Proposition 2

### Remark 5

Equations (31)–(34) are *explicit in* \(\eta \)-*direction* in the sense that they express the “unknown” quantities \({{T}}_\eta v^\epsilon \), \({{T}}_\eta w^\epsilon \), \({{T}}_y\mathfrak {k}^\epsilon \) and \({{T}}_x\mathfrak {l}^\epsilon \) in terms of the “given” eight quantities summarized in \({{T}}{\theta }^\epsilon \).

The rest of this section is devoted to the proof of Proposition 2. Since \(\epsilon >0\) is fixed in the derivation of (31)–(34), we shall omit the superscript \(\epsilon \) on the occurring quantities.

For the derivation of (31)–(34), one can obviously work locally: it suffices to fix some point in \(\varOmega ^{[xxyy]\epsilon }(r|h)\) and to consider the eight values of *v*, *w* on the midpoints of the four elementary squares incident to that vertex, and the four values of \(\mathfrak {k}\), \(\mathfrak {l}\) on the respective connecting edges.

#### 3.5.1 Derivation of Equation (31)

*v*and

*w*. Take the logarithm to obtain (31).

#### 3.5.2 Derivation of Equation (32)

*v*,

*w*) and \(({\tilde{v}},{\tilde{w}})\), so we can assume that values for \(({\tilde{v}}_0,{\tilde{w}}_0)\), \(({\tilde{v}}_L,{\tilde{w}}_L)\), \(({\tilde{v}}_R,{\tilde{w}}_R)\) are given as well. Using that \(N_R\) is the normalized cross product \(a_+\times b_0\), it is elementary to derive the following representation of \(a_+\):

*v*,

*w*) as a function of \(({\tilde{v}},{\tilde{w}})\). More precisely, by (23), one has that

*v*and

*w*approximate \({\tilde{v}}\) and \({\tilde{w}}\), respectively, to order \(\epsilon ^2\), in the sense that the family of functions

#### 3.5.3 Derivation of Equation (33)

The derivation of Eq. (34) is analogous.

## 4 The Abstract Convergence Result

In this section, we analyze the convergence of solutions to the classical Gauss-Codazzi system (9)–(12) by solutions to the discrete system (31)–(34). This is the core part of the convergence proof, from which our main result will be easily deduced in the next section.

### 4.1 Domains

A key concept in the proof is to work with analytic extensions of the quantities *v*, *w*, \(\mathfrak {k}\) and \(\mathfrak {l}\) defined in Sect. 3.4. The analytic setting forces us to introduce yet another class of domains, and corresponding spaces of real analytic functions. In the following, we assume that \(r>0\) and \(\bar{\rho }>0\) are fixed parameters (which will be frequently omitted in notations), while \(h\in (0,\bar{\rho })\) and \(\epsilon >0\) may vary, with the restriction that \(\epsilon <h\).

*analytic fattening*\(\widehat{\varOmega }_{\bar{\rho }}(r|h)\) as follows:

Replacing \(\varOmega (r|h)\) by \(\varOmega ^{[xy]\epsilon }(r|h)\) above yields definitions for analytically fattened domains \(\widehat{\varOmega }_{\bar{\rho }}^{[xy]\epsilon }(r|h)\) with respective spaces \(C^{\omega }\big (\widehat{\varOmega }^{[xy]\epsilon }(r|h)\big )\), semi-norms \(\left| \cdot \right| ^{[xy]\epsilon }_{\eta ,\rho }\) and \(\big \{\cdot \big \}^{[xy]\epsilon }_{h,\delta }\), and norms \(\left\| \cdot \right\| ^{[xy]\epsilon }_{h}\) etc.

### 4.2 Statement of the Approximation Result

Recall that \(r>0\) and \(\bar{\rho }>0\) are fixed parameters.

### Definition 7

An *analytic solution* \(\theta =(v,w,\mathfrak {k},\mathfrak {l})\) *of the classical Gauss-Codazzi system on* \(\widehat{\varOmega }_{\bar{\rho }}(r|h)\) consists of four functions \(v,w,\mathfrak {k},\mathfrak {l}\in C^{\omega }\big (\widehat{\varOmega }(r|h)\big )\) that are globally bounded on \(\widehat{\varOmega }_{\bar{\rho }}(r|h)\), are continuously differentiable with respect to \(\eta \), and satisfy Eqs. (9)–(12) on \(\widehat{\varOmega }_{\bar{\rho }}(r|h)\).

An *analytic solution* \(\theta ^\epsilon =(v^\epsilon ,w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon )\) *of the* \(\epsilon \)-*discrete Gauss-Codazzi* *system on* \(\widehat{\varOmega }_{\bar{\rho }}(r|h)\) consists of four functions \(v^\epsilon ,w^\epsilon \in C^{\omega }\big (\widehat{\varOmega }^{[xy]\epsilon }(r|h)\big )\), \(\mathfrak {k}^{\epsilon }\in C^{\omega }\big (\widehat{\varOmega }^{[xxy]\epsilon }(r|h)\big )\), \(\mathfrak {l}^\epsilon \in C^{\omega }\big (\widehat{\varOmega }^{[xyy]\epsilon }(r|h)\big )\) that satisfy Eqs. (31)–(34) on \(\widehat{\varOmega }_{\bar{\rho }}^{[xxyy]\epsilon }(r|h)\).

### Proposition 3

### Remark 6

The formulation of the proposition suggests that the height *h* of the domain on which convergence takes place is small. However, this is misleading in general. As it turns out in the proof, the limitation for *h* is mostly determined by the value of \(\bar{\rho }\). In many examples of interest, \(\bar{\rho }\) is large compared to the region of interest (determined by \(\bar{h}\) and *r*), and consequently, one has \(h^\epsilon =\bar{h}\) above, i.e., convergence takes place on the entire domain of definition of \(\theta \).

The rest of this section is devoted to the proof of Proposition 3.

### 4.3 Consistency

### Lemma 5

*G*that depends on \(\theta \), and \(\theta ^\epsilon \) only via \(\big \{\!\{\theta ^\epsilon -\theta \}\!\big \}_{h,0}\), but is independent of \(\epsilon \).

### Proof

### 4.4 Stability

*Estimate on*\(\varDelta v^\epsilon \). We begin by proving the estimate for the

*v*-component of \(\varDelta \theta ^\epsilon \). Since \(v^\epsilon \) is defined on \(\widehat{\varOmega }_{\bar{\rho }}^{[xy]\epsilon }(r|h)\), the step \(n\rightarrow n+1\) requires to estimate the values of \(\varDelta v^\epsilon (\cdot ,\eta ^*)\) for \(\eta ^*\in ((n-1)\frac{\epsilon }{2},n\frac{\epsilon }{2}]\). Choose such an \(\eta ^*\), and define accordingly \(\ell \) such that \(\eta ^*_0:=\eta ^*-\ell \epsilon \in (-\frac{\epsilon }{2},\frac{\epsilon }{2}]\); in fact, \(2\ell =n\) if

*n*is even, and \(n=2\ell +1\) if

*n*is odd. For \(0\le k\le 2\ell \), introduce

*k*are admitted. (55) is consistent with the definition of \(\eta ^*_0\), and moreover, \(\eta ^*=\eta ^*_{2\ell }\). Using the evolution equation (49), we obtain

*G*there is controlled in terms of \(\big \{\!\{\theta ^\epsilon -\theta \}\!\big \}_{n\frac{\epsilon }{2},0}\), but the induction estimate (54) is not sufficient to provide such a uniform bound, due to the weight \(\varLambda \). Fortunately, a close inspection of the terms in \(\mathrm {(III)}\) reveals in combination with (57) that we only need bounds on \(|\tilde{g}^\epsilon _j|_{\eta ,\rho }\) where \(\rho /\bar{\rho }<1-\eta ^*/h-\frac{\epsilon }{2}/h\). It is easily deduced from Lemma 5 that an \(\epsilon \)-uniform estimate on \(\big \{\!\{\theta ^\epsilon -\theta \}\!\big \}_{n\frac{\epsilon }{2},\delta }\) with \(\delta :=\bar{\rho }/h\frac{\epsilon }{2}>0\) suffices in this case, and the latter is obtained by combining (54) with (48). Enlarging

*G*if necessary, we arrive at

*Estimate on*\(\varDelta w^\epsilon \). For estimation of the

*w*-component, let \(\eta ^*\in ((n-1)\frac{\epsilon }{2},n\frac{\epsilon }{2}]\) be given as before, and define \(\eta ^*_k\) as in (55). In analogy to (56), we have

*Estimate on*\(\varDelta \mathfrak {k}^\epsilon \). Finally, let us estimate \(\varDelta \mathfrak {k}^\epsilon (\cdot ,\eta ^*)\) at some \(\eta ^*\in ((n-\frac{3}{2})\frac{\epsilon }{2},(n-\frac{1}{2})\frac{\epsilon }{2}]\). For the estimates below, let in addition a \(\xi ^*\in \mathbb {C}\) be given such that \((\xi ^*,\eta ^*)\in \widehat{\varOmega }_{\bar{\rho }}^{[xxy]\epsilon }(r|h)\). We need to use a slightly different normalization for the \(\eta ^*_k\) in (55): write \(\eta ^*=\eta ^*_{-\frac{1}{2}}+m\frac{\epsilon }{2}\) for suitable \(\eta ^*_{-\frac{1}{2}}\in (-\frac{\epsilon }{4},\frac{\epsilon }{4}]\) and a (uniquely determined) \(m\in \mathbb {N}\). Now define

*Estimate on*\(\varDelta \mathfrak {l}^\epsilon \). This is completely analogous to the estimate for \(\varDelta \mathfrak {k}^\epsilon \) above.

Summarizing the results in (60), (64) and (66), we obtain (54) with \(n+1\) in place of *n*, for an arbitrary choice of \(B>A\), and any corresponding \(h>0\) that is sufficiently small to make the coefficients in front of \(\epsilon \) in (60), (64) and (66) smaller than *B*. Notice that the implied smallness condition on *h* is independent of \(\epsilon \).

## 5 The Continuous Limit of Discrete Isothermic Surfaces

We are finally in the position to formulate and prove our main approximation result.

### 5.1 From Björling Data to Cauchy Data and Back

*f*,

*n*) in the sense of Theorem 1, first compute the associated frame \(\varPsi ^0\), the conformal factor \(u^0\), and the derived quantities \(v^0,w^0,\mathfrak {k}^0,\mathfrak {l}^0\) as functions on \((-r,r)\) as detailed in the proof there. We claim that, for any sufficiently small \(\epsilon >0\), associated Björling data \(f^\epsilon :\varOmega (r|\frac{\epsilon }{2})\rightarrow \mathbb {R}^3\) for construction of an \(\epsilon \)-discrete isothermic surface can be prescribed such that the following are true:

- (1)The initial surface piece and its tangent vectors are approximated to first order in \(\epsilon \),where the \(\mathscr {O}(\epsilon )\) indicate \(\epsilon \)-smallness that is uniform in \((\xi ,\eta )\) on the domains \(\varOmega (r|\frac{\epsilon }{2})\) for \(f^\epsilon \), and \(\varOmega ^{[x]\epsilon }(r|\frac{\epsilon }{2})\) for \(\delta _xf^\epsilon \), and \(\varOmega ^{[y]\epsilon }(r|\frac{\epsilon }{2})\) for \(\delta _yf^\epsilon \), respectively.$$\begin{aligned} f^\epsilon (\xi ,\eta )&= f(\xi ) + \mathscr {O}(\epsilon ),\nonumber \\ \delta _xf^\epsilon (\xi ,\eta )&= \exp (u^0(\xi ))\varPsi ^0_1(\xi ) + \mathscr {O}(\epsilon ),\nonumber \\ \delta _yf^\epsilon (\xi ,\eta )&= \exp (u^0(\xi ))\varPsi ^0_2(\xi ) + \mathscr {O}(\epsilon ), \end{aligned}$$(67)
- (2)The derived quantities \((v^\epsilon , w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon )\) satisfyat each point \((\xi ,\eta )\) in \(\varOmega ^{[xy]\epsilon }(r|\epsilon )\) for \(v^\epsilon ,{\tilde{w}}^\epsilon \), in \(\varOmega ^{[xxy]\epsilon }(r|\epsilon )\) for \(\mathfrak {k}^\epsilon \), and in \(\varOmega ^{[xyy]\epsilon }(r|\epsilon )\) for \(\mathfrak {l}^\epsilon \), respectively.$$\begin{aligned} v^\epsilon (\xi ,\eta )=v^0(\xi ), \quad {\tilde{w}}^\epsilon (\xi ,\eta )=w^0(\xi ), \quad \mathfrak {k}^\epsilon (\xi ,\eta )=\mathfrak {k}^0(\xi ), \quad \mathfrak {l}^\epsilon (\xi ,\eta )=\mathfrak {l}^0(\xi ), \end{aligned}$$(68)

Notice that the data \((v^\epsilon , w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon )\) are \(\xi \)-analytic quantities; ironically, one cannot even expect continuity of the respective data \(f^\epsilon \) in general.

- (1)
*a*is parallel to \(\varPsi ^0_1(0)\), and*b*is orthogonal to*n*(0), - (2)
\(\displaystyle {\epsilon w^0(\xi ) = \frac{\langle a,b\rangle }{\Vert a\Vert \Vert b\Vert }}\),

- (3)
\(\Vert a\Vert =\exp \big (u^0(0)+\frac{\epsilon }{2}v^0(\xi )\big )\) and \(\Vert b\Vert =\exp \big (u^0(0)-\frac{\epsilon }{2}v^0(\xi )\big )\).

If instead \(-\frac{\epsilon }{2}<\eta \le 0\), then we define \(f^\epsilon (\xi ,\eta +\frac{\epsilon }{2})=f(0)\), and we assign data \(f^\epsilon \) at \((\xi +\frac{\epsilon }{2},\eta )\) and at \((\xi -\frac{\epsilon }{2},\eta )\) with the respective adaptations for the conditions on the vectors.

- (1)
the \(\sin \)-value of the angle between the planes spanned by (

*a*,*b*) and by (*b*,*c*), respectively, equals to \(\epsilon \mathfrak {k}^0(\xi -3\frac{\epsilon }{4})\), - (2)
\(\displaystyle {\frac{\langle a,c\rangle }{\Vert a\Vert \Vert c\Vert }=\epsilon w^0(\xi -\frac{\epsilon }{2})}\),

- (3)
\(\Vert c\Vert =\Vert b\Vert \exp (\frac{\epsilon }{2}v^0(\xi -\frac{\epsilon }{2}))\).

By continuing this construction in an inductive manner, we enlarge the domain of definition with respect to \(\xi \) by \(\frac{\epsilon }{2}\) in both directions in each step, until \(f^\epsilon \) is defined on all of \(\varOmega (r|\frac{\epsilon }{2})\). It is obvious from the construction that (68) holds. The verification of (67) is a tedious but straight-forward exercise in elementary geometry that we leave to the interested reader. An important point is that the aforementioned construction only uses data that can be obtained very directly from the Björling data (*f*, *n*). Indeed, the calculation of \(u^0\), \(\varPsi ^0\) and \((v^0,w^0,\mathfrak {k}^0,\mathfrak {l}^0)\) from (*f*, *n*) only involves differentiation and inversion of matrices. In particular, all operations are local.

### Definition 8

Assume that analytic Björling data (*f*, *n*) and discrete data \(f^\epsilon :\varOmega (r|\frac{\epsilon }{2})\) are given such that (67) and (68) are satisfied. The maximal \(\epsilon \)-discrete isothermic surface \(F^\epsilon :\varOmega (r|h_\epsilon )\) that is obtained from \(f^\epsilon \) as Björling data—see Proposition 1—is referred to as *grown from* (*f*, *n*).

### 5.2 Main Result

The central approximation result is the following.

### Theorem 2

Let analytic Björling data (*f*, *n*) on \((-r,r)\) be given, and let \(F:\varOmega (r|\bar{h})\rightarrow \mathbb {R}^3\) be the corresponding real-analytic isothermic surface. Further, let \(F^\epsilon :\varOmega (r|h^\epsilon )\rightarrow \mathbb {R}^3\) be the family of \(\epsilon \)-discrete isothermic surfaces that are grown from (*f*, *n*).

*F*that is determined by given Björling data by a family of \(\epsilon \)-isothermic surfaces \(F^\epsilon \). Our construction of the surface \(F^\epsilon \) is completely explicit, and it requires no a priori knowledge about

*F*. The plots in Figs. 5 and 6 illustrate that our construction can be used to generate pictures of the surfaces \(F^\epsilon \) with just a few lines of code.

### Remark 7

Note that if the discrete isothermic surfaces \(F^\epsilon \) converge to a smooth isothermic surface *F*, then also the discrete Christoffel and Darboux transforms of \(F^\epsilon \) converge to the corresponding smooth Christoffel and Darboux transforms of *F*. This will be proven in Sect. 6.

### Proof

(of Theorem 2) Since *F* is real-analytic on \(\varOmega (r|\bar{h})\), the derived quantities *u*, *v*, *w* and \(\mathfrak {k},\mathfrak {l}\) are real-analytic there as well, and can be extended to functions in \(C^{\omega }\big (\widehat{\varOmega }(r|\bar{h})\big )\), for a suitable choice of the “fattening parameter” \(\bar{\rho }>0\), after diminishing \(\bar{h}>0\) if necessary. The extensions satisfy the Gauss-Codazzi system (9)–(12) on the complexified domain.

Next, consider the \(\epsilon \)-discrete isothermic surfaces \(F^\epsilon :\varOmega (r|h_\epsilon )\rightarrow \mathbb {R}\) that are grown from the Björling data (*f*, *n*). Define the associated quantities \(v^\epsilon ,w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon \). Thanks to (68), these are real analytic functions on \(\varOmega ^{[xy]\epsilon }(r|\epsilon )\), and they extend complex-analytically w.r.t. \(\xi \) to \(\widehat{\varOmega }_{\bar{\rho }}^{[xy]\epsilon }(r|\epsilon )\). Since the quadruple \((v^\epsilon ,w^\epsilon ,\mathfrak {k}^\epsilon ,\mathfrak {l}^\epsilon )\) satisfies the discrete Gauss-Codazzi equations (31)–(34), the \(\xi \)-analyticity is propagated from the initial strip to the maximal domain of existence, i.e., \(v^\epsilon ,w^\epsilon \in C^{\omega }\big (\widehat{\varOmega }^{[xy]\epsilon }(r|h_\epsilon )\big )\) etc.

*A*. For the remaining differences \(\varDelta w^\epsilon \), it follows via Lemma 3 on the equivalence of \((v^\epsilon ,w^\epsilon )\) and \(({\tilde{v}}^\epsilon ,{\tilde{w}}^\epsilon )\) that they satisfy the same estimate (enlarging

*A*if necessary):

We only sketch the proof of (71), that is little more than a repeated application of the Gronwall lemma. For further technical details, we refer the reader to [2], where the relevant estimates have been carried out in a very similar situation, see the proof of Theorem 5.4 therein.

A posteriori, we conclude that \(h_\epsilon \uparrow h\) as \(\epsilon \downarrow 0\). where \(h>0\) is the constant obtained in the proof of Proposition 3. Indeed, thanks to the uniform closeness of the discrete tangent vectors \(\delta _xF^\epsilon \), \(\delta _yF^\epsilon \) to their continuous counterparts \(\partial _xF\), \(\partial _yF\)—which are orthogonal with non-vanishing length—it easily follows that there cannot occur any degeneracies in \(F^\epsilon \) at any \(h^\epsilon <h\). \(\square \)

## 6 Transformations

Isothermic surfaces have an exceptionally rich transformation theory. For the definition of discrete isothermic surfaces used in this paper this transformation theory carries over to the discrete setup.

We consider two important transformations, namely the Christoffel transformation and Darboux transformation. Their analogs for discrete isothermic surfaces may for example be found in [3, 4, 14, 15, 16]. It is a natural question whether the convergence results of Theorem 2 can be generalized to imply the convergence of the transformed surfaces.

### 6.1 Christoffel Transformation

We briefly remind the classical definition of the Christoffel transformation. The included existence claim was first proved by Christoffel [10].

### Definition 9

*dual*to the surface

*F*or

*Christoffel transform*of the surface

*F*.

*F*and its dual \(F^\star \), straightforward calculation leads to the following relations between corresponding quantities.

### Definition 10

*dual*to \(F^\epsilon \) or

*Christoffel transform*of \(F^\epsilon \).

### Corollary 1

Under the assumptions of Theorem 2 not only the discrete isothermic surface itself converges to the corresponding smooth isothermic surface, but also the discrete Christoffel transforms converge to the corresponding Christoffel transforms of the smooth isothermic surface.

### Proof

### 6.2 Darboux Transformation

The Darboux transformation for isothermic surfaces was introduced by Darboux [12]. It is a special case of a Ribaucour transformation and is closely connected to Möbius geometry as well as to the integrable system approach to isothermic surfaces, see for example [15].

### Definition 11

*Darboux transform*of

*F*if

*parameter of the Darboux transformation*.

### Definition 12

Let \(F^\epsilon :\varOmega (r|h)\rightarrow \mathbb {R}^3\) be a discrete isothermic surface. Then the discrete isothermic surface \((F^+)^\epsilon :\varOmega (r|h)\rightarrow \mathbb {R}^3\) is called a *discrete Darboux transform* of \(F^\epsilon \) if the following conditions are satisfied.

- (i)
The four points \({{T}}^{-1}_x F^\epsilon , {{T}}_x F^\epsilon , {{T}}^{-1}_x (F^+)^\epsilon ,{{T}}_x (F^+)^\epsilon \) lie in a common plane and the same is true for \({{T}}^{-1}_y F^\epsilon \), \({{T}}_y F^\epsilon \), \({{T}}^{-1}_y (F^+)^\epsilon \), \({{T}}_y (F^+)^\epsilon \).

- (ii)
\(\displaystyle q({{T}}^{-1}_x F^\epsilon , {{T}}_x F^\epsilon , {{T}}_x (F^+)^\epsilon , {{T}}^{-1}_x (F^+)^\epsilon )=\frac{1}{\gamma }\) and \(\displaystyle q({{T}}^{-1}_y F^\epsilon ,{{T}}_y F^\epsilon ,{{T}}_y (F^+)^\epsilon , {{T}}^{-1}_y(F^+)^\epsilon )=-\frac{1}{\gamma },\) where \(\gamma \in \mathbb {R}\), \(\gamma \ne 0\) is a constant which is called

*parameter of the Darboux transformation*.

Note that given any discrete isothermic surface, a discrete Darboux transform may be obtained by prescribing the value of \((F^+)^\epsilon \) at one point and using the conditions of the definition to successively build a new surface which is also discrete isothermic (as long as the surface does not degenerate).

In order to obtain convergence of the discrete Darboux transform to the corresponding continuous one, we choose \(\gamma =C/\epsilon ^2\).

### Corollary 2

Under the assumptions of Theorem 2 not only the discrete isothermic surface itself converges to the corresponding smooth isothermic surface, but also the discrete Darboux transforms (with \(\gamma =C/\epsilon ^2\)) converge to the corresponding Darboux transforms (with parameter *C*) of the smooth isothermic surface.

### Proof

Assume that the discrete isothermic surface itself converges to the corresponding smooth isothermic surface with errors of order \(\mathscr {O}(\epsilon )\) as in the proofs of Theorem 2. Now start with \((F^+)^\epsilon (0,0)=F^+(0,0)\) and build the discrete Darboux transform successively using the above definition. Denote the distance between corresponding points by \(d^\epsilon =(F^+)^\epsilon -F^\epsilon \).

### Remark 8

Using the definitions of continuous and discrete Darboux transformations, corresponding formulas for \(a^+\), \(b^+\), \({\hat{u}}^+\), \({\check{u}}^+\), \(N^+\), \({\tilde{v}}^+\), \({\tilde{w}}^+\), \(v^+\), \(w^+\), \(\mathfrak {k}^+\), \(\mathfrak {l}^+\) may be deduced, which also converge under the assumptions of Theorem 2.

## Footnotes

- 1.
The system of Gauss-Codazzi-equations can be simplified to Calapso’s equation [8], which is a single scalar fourth order PDE, but unfortunately of indefinite type.

- 2.
Cyclic ordering means that walking around the circle either clockwise or anti-clockwise, one passes \(p_1\), \(p_2\), \(p_3\) and \(p_4\) in that order, see Fig. 3

*(left)*.

## Notes

### Acknowledgments

The authors would like to thank Sepp Dorfmeister and Fran Burstall for stimulating and helpful discussions. We further thank the anonymous referee for the careful reading of the initial manuscript and various suggestions for improvement. This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

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