Advances in Discrete Differential Geometry pp 241-265 | Cite as

# Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

## Abstract

Given a triangulated region in the complex plane, a discrete vector field *Y* assigns a vector \(Y_i\in \mathbb {C}\) to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

## 1 Introduction

*U*in the complex plane \(\mathbb {C}\cong \mathbb {R}^2\) with coordinates \(z=x+iy\) together with a holomorphic vector field

*Y*is a real vector field. It assigns to each \(p\in \mathbb {R}^2\) the vector \(f(p)\in \mathbb {C}\cong \mathbb {R}^2\). We do not consider objects like \(\frac{\partial }{\partial z}\) which are sections of the complexified tangent bundle \(({{\mathrm{T}}}\mathbb {R}^2)^{\mathbb {C}}\).

*Y*(defined for small

*t*on open subsets of

*U*with compact closure in

*U*). Then the euclidean metric pulled back under \(g_t\) is conformally equivalently to the original metric:

*u*. The infinitesimal change in scale \({\dot{u}}\) is given by

*z*yields one half the third derivative of

*f*:

*Y*corresponds to an infinitesimal Möbius transformation of the extended complex plane \(\overline{\mathbb {C}}\) if and only if

*f*is a quadratic polynomial. In this sense \(f_{zzz}\) measures the infinitesimal “change in Möbius structure” under

*Y*(Möbius structures are sometimes also called “complex projective structures” [6]). Moreover, the holomorphic quadratic differential

*q*is unchanged under a change of variable \(\varPhi (z)=w=\xi +i\eta \) whenever \(\varPhi \) is a Möbius transformation. This is easy to see if \(\varPhi (z)=az+b\) is an affine transformation. In this case

For realizations from an open subset *U* of the Riemann sphere \(\mathbb {C}\text {P}^1\) the vanishing of the Schwarzian derivative characterizes Möbius transformations. The quadratic differential *q* plays a similar role for vector fields. We call *q* the *Möbius derivative* of *Y*.

An important geometric context where holomorphic quadratic differentials arise comes from the theory of minimal surfaces: Given a simply connected Riemann surface *M* together with a holomorphic immersion \(g:M \rightarrow S^2 \subset \mathbb {R}^3\) and a holomorphic quadratic differential *q* on *M*, there is a minimal surface \(F:M\rightarrow \mathbb {R}^3\) (unique up to translations) whose Gauß map is *g* and whose second fundamental form is \(\text{ Re }\,q\).

In this paper we will provide a discrete version for all details of the above story. Instead of smooth surfaces we will work with triangulated surfaces of arbitrary combinatorics. The notion of conformality will be that of conformal equivalence as explained in [3]. Holomorphic vector fields will be defined as infinitesimal conformal deformations.

There is also a completely parallel discrete story where conformal equivalence of planar triangulations is replaced by preserving intersection angles of circumcircles. To some extent we also tell this parallel story that belongs to the world of circle patterns.

The results on planar triangular meshes in this paper are closely related to isothermic triangulated surfaces in Euclidean space [8].

## 2 Discrete Conformality

In this section, we review two notions of discrete conformality for planar triangular meshes. We first start with some notations of triangular meshes.

### Definition 2.1

A triangular mesh *M* is a simplicial complex whose underlying topological space is a connected 2-manifold (with boundary). The set of vertices (0-cells), edges (1-cells) and triangles (2-cells) are denoted as *V*, *E* and *F*.

We denote \(E_{int}\) the set of interior edges and \(V_{int}\) the set of interior vertices. Without further notice we will assume that all triangular meshes under consideration are oriented.

### Definition 2.2

A *realization* \(z:V \rightarrow \mathbb {C}\) of a triangular mesh *M* in the extended complex plane assigns to each vertex \(i\in V\) a point \(z_i \in \overline{\mathbb {C}}\) in such a way that for each triangle \(\{i\!jk\}\in F\) the points corresponding to its three vertices are not collinear.

*z*. In particular, we want

*z*to be conformally equivalent to \(g\circ z\) whenever \(g: \overline{\mathbb {C}} \rightarrow \overline{\mathbb {C}}\) is a Möbius transformations. This requirement will certainly be met if we base our definitions on complex cross ratios: Given a triangular mesh \(z:V\rightarrow \mathbb {C}\), we associate a complex number to each interior edge \(\{i\!j\} \in E_{int}\), namely the

*cross ratio*of the corresponding four vertices (See Fig. 1)

*conformal equivalence theory*[9, 13] and

*circle pattern theory*[11].

Note that for the sake of simplicity of exposition we are ignoring here realizations in \(\overline{\mathbb {C}}\) where one of the vertices is mapped to infinity.

### 2.1 Conformal Equivalence

The edge lengths of a triangular mesh realized in the complex plane provide a discrete counterpart for the induced Euclidean metric in the smooth theory. A notion of conformal equivalence based on edge lengths was proposed by Luo [9]. Later Bobenko et al. [3] stated this notion in the following form:

### Definition 2.3

*conformally equivalent*if the norm of the corresponding cross ratios are equal:

This definition can be restated in an equivalent form that closely mirrors the notion of conformal equivalence of Riemannian metrics:

### Theorem 2.4

### Proof

*u*implies conformal equivalence. Conversely, for two conformally equivalent realizations

*z*,

*w*, we define a function \(\sigma : E \rightarrow \mathbb {R}\) by

*z*,

*w*are conformally equivalent \(\sigma \) satisfies for each interior edge \(\{i\!j\}\)

*i*and any triangle \(\{i\!jk\}\) containing it we then define

*i*is a triangulated disk if

*i*is interior, or is a fan if

*i*is a boundary vertex. Hence the value \(u_i\) defined in this way is independent of the chosen triangle. \(\square \)

### 2.2 Circle Patterns

Based on these angles we obtain another notion of discrete conformality which reflects the angle-preserving property that we have in the smooth theory.

### Definition 2.5

*pattern structure*if the corresponding intersection angles of neighboring circumscribed circles are equal:

Just as conformal equivalence was related to scale factors *u* at vertices, having the same pattern structure is related to the existence of certain angular velocities \(\alpha \) located at vertices:

### Theorem 2.6

### Proof

*i*and any triangle \(\{i\!jk\}\) containing it we define \(\alpha _i \in [0, 2\pi )\) such that

*i*is a triangulated disk if

*i*is interior, or is a fan if

*i*is a boundary vertex. Hence having the same pattern structure implies that the value \(\alpha _i\) is independent of the chosen triangle. \(\square \)

## 3 Infinitesimal Deformations and Linear Conformal Theory

We will linearize both of the above notions of discrete conformality by considering infinitesimal deformations. This will allow us to relate them to linear discrete complex analysis, based on a discrete analogue of the Cauchy Riemann equations [4, 5, 10] (See the survey [12]).

### Definition 3.1

*infinitesimal conformal deformation*of a realization \(z:V \rightarrow \mathbb {C}\) of a triangular mesh is a map \({\dot{z}}: V \rightarrow \mathbb {C}\) such that there exists \(u:V \rightarrow \mathbb {R}\) satisfying

*u*the

*scale change*at vertices.

### Definition 3.2

*infinitesimal pattern deformation*of a realization \(z:V \rightarrow \mathbb {C}\) of a triangular mesh is a map \({\dot{z}}: V \rightarrow \mathbb {C}\) such that there exists \(\alpha :V \rightarrow \mathbb {R}\) satisfying

*angular velocities*at vertices.

### Example 3.3

Infinitesimal conformal deformations and infinitesimal pattern deformations are closely related:

### Theorem 3.4

Suppose \(z:V \rightarrow \mathbb {C}\) is a realization of a triangular mesh. Then an infinitesimal deformation \({\dot{z}}:V \rightarrow \mathbb {C}\) is conformal if and only if \(i {\dot{z}}\) is a pattern deformation.

### Proof

### 3.1 Infinitesimal Deformations of a Triangle

Let \(z:V \rightarrow \mathbb {C}\) be a realization of a triangulated mesh and \({\dot{z}}\) an infinitesimal deformation. Up to an infinitesimal translation \({\dot{z}}\) is completely determined by the infinitesimal scalings and rotations that it induces on each edge. These infinitesimal scalings and rotations of edges satisfy certain compatibility conditions on each triangle. These conditions involve the cotangent coefficients well known from the theory of discrete Laplacians. As we will see in Sect. 3.2, for conformal deformations (as well as for pattern deformations) the infinitesimal scalings and rotations of edges are indeed discrete harmonic functions.

*i*,

*j*,

*k*denotes any cyclic permutation of the indexes 1, 2, 3. The triangle angle at the vertex

*i*is denoted by \(\beta _i\). We adopt the convention that all \(\beta _1,\beta _2,\beta _3\) have positive sign if the triangle \(z_1,z_2,z_3\) is positively oriented and a negative sign otherwise. Suppose we have an infinitesimal deformation of this triangle. Then there exists \(\sigma _{i\!j},\omega _{i\!j} \in \mathbb {R}\) such that

### Lemma 3.5

Given \(\sigma _{i\!j},\omega _{i\!j} \in \mathbb {R}\) the following statements are equivalent:

(a) There exist \({\dot{z}}_i\) such that (2) holds.

### Proof

*A*denoting the signed triangle area we have the following identities:

*R*denotes the circumradius of the triangle. Thus \(\sigma \) signifies an average scaling of the triangle.

### 3.2 Harmonic Functions with Respect to the Cotangent Laplacian

In smooth complex analysis conformal maps are closely related to harmonic functions. If a conformal map preserves orientation it is holomorphic and satisfies the Cauchy Riemann equations. In particular, its real part and the imaginary part are conjugate harmonic functions. Conversely, given a harmonic function on a simply connected surface then it is the real part of some conformal map.

A similar relationship manifests between discrete harmonic functions (in the sense of the cotangent Laplacian) and infinitesimal deformations of triangular meshes. Discrete harmonic functions can be regarded as the real part of holomorphic functions which satisfies a discrete analogue of the Cauchy Riemann equations. In particular, a relation between discrete harmonic functions and infinitesimal pattern deformations was found by Bobenko, Mercat and Suris [2]. Integrable systems were involved in this context. We extend their result to include the case of infinitesimal conformal deformations.

### Theorem 3.6

Let \(z:V \rightarrow \mathbb {C}\) be a simply connected triangular mesh realized in the complex plane and \(h: V \rightarrow \mathbb {R}\) be a function. Then the following are equivalent:

*h*is a harmonic function with respect to the cotangent Laplacian, i.e. using the notation of Fig. 1, for all interior vertices \(i\in V_{int}\) we have

*h*. It is unique up to infinitesimal rotations and translations.

(c) There exists an infinitesimal pattern deformation \(i{\dot{z}}:V \rightarrow \mathbb {C}\) with *h* as angular velocities. It is unique up to infinitesimal scalings and translations.

### Proof

We show the equivalence of the first two statements. The equivalence of the first and the third follows similarly.

*h*is a harmonic function. Since the triangular mesh is simply connected, equation (5) implies the existence of a function \(\tilde{\omega }: F \rightarrow \mathbb {R}\) such that for all interior edges \(\{i\!j\}\) we have

*conjugate harmonic function*of

*h*. Using \(\tilde{\omega }\) we define a function \(\omega :E \rightarrow \mathbb {R}\) via

*z*with

*h*as scale factors.

To show uniqueness, suppose \({\dot{z}}, {\dot{z}}'\) are infinitesimal conformal deformations with the same scale factors. Then \({\dot{z}} - {\dot{z}}'\) preserves all the edge lengths of the triangular mesh and hence is induced from an Euclidean transformation.

*h*. We write

*h*is harmonic. \(\square \)

## 4 Holomorphic Quadratic Differentials

In this section, we introduce a discrete analogue of holomorphic quadratic differentials. We illustrate their correspondence to discrete harmonic functions. It reflects the property in the smooth theory that holomorphic quadratic differentials parametrize Möbius structures on Riemann surfaces ([6, Chap. 9]).

*M*, a complex-valued function \(\eta :\vec {E}\rightarrow \mathbb {C}\) is called a

*discrete 1-form*if

*closed*if for every face \(\{i\!jk\}\)

*exact*if there exists a function \(f:V \rightarrow \mathbb {C}\) such that

*M*and these are called

*dual 1-forms*. Given an oriented edge

*e*, we denote \(e^*\) its dual edge oriented from the right face of

*e*to its left face. The set of oriented dual edges is denoted by \(\vec {E}^*\).

### Definition 4.1

*discrete holomorphic quadratic differential*if it satisfies for every interior vertex \(i \in V_{int}\)

### Theorem 4.2

Let \(q:E_{int} \rightarrow i\mathbb {R}\) be a holomorphic quadratic differential on a realization \(z:V \rightarrow \mathbb {C}\) of a triangular mesh. Suppose \(\varPhi :\overline{\mathbb {C}} \rightarrow \overline{\mathbb {C}}\) is a Möbius transformation which does not map any vertex to infinity. Then *q* is again a holomorphic quadratic differential on \(w:= \varPhi \circ z\).

### Proof

We are going to show that on a simply connected triangular mesh, there is a correspondence between discrete holomorphic quadratic differentials and discrete harmonic functions.

*M*. If we interpolate it piecewise-linearly over each triangular face, its gradient is constant on each face and we have \({{\mathrm{grad}}}_z u:F \rightarrow \mathbb {C}\) given by

*z*in \(\mathbb {C}\) in an orientation reversing fashion the area \(A_{i\!jk}\) is considered to have a negative sign. Granted this, one can verify that the gradient of

*u*satisfies

*M*by

### Lemma 4.3

### Proof

### Lemma 4.4

### Proof

*q*is well defined since

*u*is harmonic if and only if

*q*is a holomorphic quadratic differential. \(\square \)

### Lemma 4.5

### Proof

*M*defined by

*M*is simply connected and

### Theorem 4.6

Furthermore, the space of holomorphic quadratic differentials is a vector space isomorphic to the space of discrete harmonic functions module linear functions.

### Proof

In previous sections, we showed that every harmonic function corresponds to an infinitesimal conformal deformation. The following shows that discrete holomorphic quadratic differentials are the change in the intersection angles of circumscribed circles.

### Theorem 4.7

*u*as scale factors. Then we have

### Proof

## 5 Conformal Deformations in Terms of End(\(\mathbb {C}^2\))

In this section we show how an infinitesimal conformal deformation gives rise to a discrete analogue of a holomorphic null curve in \(\mathbb {C}^3\). Later we will see that the real parts of such a “holomorphic null curve” can be regarded as the Weierstrass representation of a discrete minimal surface.

*multiplicative dual 1-form*defined as

*i*we have

### Lemma 5.1

### Proof

We now can summarize the information about finite deformations of a realization as follows:

### Theorem 5.2

*i*

*G*. Then

Suppose we have a family of deformations described by dual 1-forms \(G_t: \vec {E}_{int} \rightarrow \text{ SL }(2,\mathbb {C})\) with \(G_0 \equiv I\). By considering \(\eta := \frac{d}{dt}|_{t=0}\, G_t\) we obtain the following description of infinitesimal deformations:

### Corollary 5.3

*i*

Note that given a mesh, the 1-form \(\eta \) is uniquely determined by the eigenfunction \(\mu \). We now investigate the constraints on \(\mu \) implied by the closedness condition (7) of \(\eta \).

## 6 Weierstrass Representation of Discrete Minimal Surfaces

The Weierstrass representation for minimal surfaces in \(\mathbb {R}^3\) is the most classical example for applications of complex analysis:

### Theorem 6.1

*n*is the stereographic projection of

*g*

*f*and encodes its second fundamental form: The direction defined by a nonzero tangent vector

*W*is

We now develop a discrete version of this theorem for arbitrary triangular meshes realized in the complex plane. A similar formula for quadrilateral meshes with factorized real cross ratios was established by Bobenko and Pinkall [1]. Here we will use the definition of a discrete minimal surface *f* with Gauß map *n* given in [8]:

### Definition 6.2

*F*of faces is called a discrete minimal surface with Gauß map

*n*if for all oriented interior edges \(e_{i\!j}\) we have

This definition mirrors the fact from the smooth theory that minimal surfaces are Christoffel duals of their Gauß maps (Fig. 3). The correspondence between discrete harmonic functions and discrete minimal surfaces was observed in [8]. Here is a Weierstrass representation for discrete minimal surfaces in terms of their Gauß map and their Hopf differential:

### Theorem 6.3

*f*is a discrete minimal surface with Gauß map

### Proof

*i*. Therefore, since the triangular mesh is simply connected, there exists \(\mathfrak {F}:F \rightarrow \mathbb {C}^3\) such that for any interior edge

*e*we have

*f*is a discrete minimal surface we define a function \(k:E_{int} \rightarrow \mathbb {R}\) by

*f*is a discrete minimal surface with Gauß map

*n*. The converse is straightforward: Given a discrete minimal surface

*f*with Gauß map

*n*we define \(k:E_{int} \rightarrow \mathbb {R}\) via (11). Then it can be shown that the function

### Remark 6.4

The discrete minimal surfaces given by (10) are trivalent meshes with planar vertex stars for purely imaginary *q*. It is closely related to discrete asymptotic nets. The factor *i* in front of \(z_{\!j}-z_i\) appears since the integration is taken over a dual mesh while in the smooth theory \(*dz=idz\).

## Notes

### Acknowledgments

This research was supported by DFG SFB/TRR 109 “Discretization in Geometry and Dynamics”

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