## Abstract

Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

## 1. Introduction

The subject of potential theory on surfaces is of interest across many areas in the mathematical sciences both from a purely abstract perspective and in the context of numerous diverse physical applications. Of fundamental importance to this subject is Green's function for the Laplace–Beltrami operator on the surface. For both analytical and computational purposes, it is valuable to have explicit representations for this function.

The simplest case is that of the Euclidean plane, and for this, Green's function is of course elementary. Green's function is also well-known for the case of the sphere; explicit representations are presented, for example, in Kimura & Okamoto [1], Kimura [2], and also Crowdy & Cloke [3]. Kimura [2] also considers the case of the hyperbolic plane. However, for more complex surfaces, for example of higher genus or non-constant curvature, the theory is mathematically more complicated and far fewer explicit results are known. Perhaps the simplest such surface is that of the ring torus, and this is therefore a natural first choice on which to attempt to extend the existing theory.

In this paper, we construct an explicit representation for Green's function for the Laplace–Beltrami operator on a toroidal surface. This representation is as a function of a single complex variable. To the best of the authors' knowledge, this is the first such explicit formula to be constructed for this function.

We have chosen deliberately to address this issue from the most general potential theoretical viewpoint, with the intention that the mathematics developed herein can be readily applied elsewhere to solve problems on the torus for which the Laplace–Beltrami equation turns out to be the governing equation. Understanding potential theory on surfaces such as the torus could yield valuable insight into interesting phenomena occurring in numerous areas, including, for example, vortex dynamics. Indeed, it is primarily within this context that [1–[3] all arose. Much of the existing theory regarding vortex dynamics on curved surfaces pertains to compact surfaces of genus zero and in particular the sphere, owing largely to the desire to understand various physical phenomena that occur on the Earth's surface; indeed, a vast body of work has emerged since the late 1970s involving various models of systems of point vortices on the sphere beginning with Bogomolov [4], and including Crowdy & Cloke [3], Kimura & Okamoto [1], Boatto & Simó [5] and Kidambi & Newton [6,7]. However, fewer results are known explicitly for more general surfaces. In addition to Kimura [2], Kim [8] has analysed the case of the spheroid, while both Hally [9] and Boatto & Koiller [10] have considered point vortex motion on general curved surfaces from a more abstract point of view. As stated in [2], results concerning more general curved surfaces are likely to find relevance beyond the realms of classical fluid mechanics. We mention, for example, quantum mechanics and flows of superfluid films [11]. Indeed, Corrada–Emmanuel [12] has found an exact solution for the velocity field for so-called superfluid film vortices on the surface of a torus. Another potential application of the results in this paper is in the so-called ‘best-packing’ problem for points on a compact surface; this problem arises in numerous areas, including statistical sampling and computer-aided design. Hardin & Saff [13] have analysed this problem on the torus, in addition to other compact surfaces. Recently, Newton & Sakajo [14] have investigated the link between these optimum point distributions and point vortex equilibria for the case of the sphere. It may be possible to conduct an analysis similar to that of Newton & Sakajo [14] on a toroidal surface.

Our method can be summarized as follows. At the heart of the approach used by Crowdy & Cloke [3] is the stereographic projection of the sphere onto the complex plane. Other authors including [2,7,8] also use stereographic projection from the surface in consideration to the complex plane. Being cast in terms of a single complex variable turns out to be a particularly expedient approach, in addition to being mathematically elegant. Indeed, viewing the surface of the sphere in this way, Crowdy and co-workers have been able to discover classes of analytical solutions describing various vortical structures on the sphere (for example, in addition to [3], see [15–17]). In the light of these advantages, we have chosen to couch our work in a similar complex variable framework. We introduce a stereographic projection of the surface of the torus onto a concentric annulus in the plane, and show how to write the Laplace–Beltrami operator on the torus in terms of the complex coordinates in this plane, and then integrate a corresponding Laplace–Beltrami equation to obtain Green's function associated with the torus. We remark that Corrada-Emmanuel [12] also employs the technique of stereographic projection to map the torus surface onto a planar rectangular cell.

In terms of our complex variable, Green's function must be doubly periodic. Similar properties are exhibited by Green's functions in doubly periodic lattices of parallelograms or rectangles in the plane [18,19] and also arise in studying point vortex dynamics in such domains [20–23]; these rectangular cells are in fact often dubbed ‘flat tori’. Solutions for these are commonly constructed in terms of elliptic functions in order to capture their required double periodicity. More recently, Crowdy [24] has dealt with such a problem by transforming the rectangular cells to concentric annuli and by using a special function, related to elliptic functions, known as the Schottky–Klein prime function. We will use this function to construct our doubly periodic solution for Green's function associated with the torus. We also appeal to another special function known as the dilogarithm function [25]. It is important to bear in mind that although we address the present problem in terms of a planar variable, the intrinsic curvature effects of the three-dimensional toroidal surface are fully incorporated into our theoretical framework.

To complete the paper, we choose to illustrate an application of our theory from a perspective in vortex dynamics, motivated by extending the existing work in this area on the sphere referred to above. In this context, Green's function is equivalent to the streamfunction for the flow. The representation we derive for this function can be viewed as the analogue on the torus of that presented by Crowdy & Cloke [3] on the sphere.

## 2. Formulation of problem

Let denote the surface of the ring torus consisting of points where
2.1Here, *R* and *r* are the major and minor radii, respectively, *R*>*r*, and *θ*,*ϕ*∈[0,2*π*]. is formed by taking a circle *C* of radius *r* centred a distance *R* from the origin in the (*x*,*z*)-plane, and rotating it through 2*π* about the *z*-axis. *θ* denotes the angle around the circle *C*, and *ϕ* denotes the azimuthal angle about the *z*-axis. It can be shown that the Laplace–Beltrami operator on the surface of the torus in terms of the coordinates (2.1) is given by
2.2We seek Green's function *ψ*(*θ*,*ϕ*) for the operator . This is a real-valued function and satisfies the equation
2.3where *δ*(*θ*,*ϕ*;*θ*_{0},*ϕ*_{0}) represents the Dirac delta function. We remark that the additive constant term on the right-hand side of (2.3) is required by Gauss's divergence theorem, and that 4*π*^{2}*rR* is the surface area of the torus . Note that we could have chosen any function to add to the right-hand side of (2.3) provided Gauss's divergence theorem is satisfied, but our choice of the constant −1/4*π*^{2}*rR* is the simplest and turns out to be meaningful in numerous physical problems where this Green's function arises.

We shall solve for *ψ*(*θ*,*ϕ*) by transforming the problem to a complex plane, as follows.

## 3. Stereographic projection

The stereographic projection from the torus to a rectangle in a complex *Z*-plane is presented in Akhiezer [26]:
3.1This map is one–one and conformal, and the dimensions of the rectangle are 2*π* and *L*, where
3.2It is well known that the conformal map from this rectangle to a concentric annulus *D*_{ζ}={*ζ*|*ρ*≤|*ζ*|≤1}, where *ρ*=e^{−L} is
3.3Thus, the stereographic projection of to *D*_{ζ} can be written in the form
3.4where
3.5and
3.6Equation (3.4) is analogous in functional form to the stereographic projection to the plane of the sphere considered by Crowdy & Cloke [3]. Note that *ζ*↦e^{2πi}*ζ* corresponds to a rotation through 2*π* in the *ϕ*-direction on , while *ζ*↦*ρζ* corresponds to a rotation through 2*π* in the *θ*-direction on . Using the facts that
3.7the operator in (2.2) can be expressed in terms of the complex variables *ζ* and :
3.8It remains to write in terms of *ζ* and . From (3.4) and (3.6), we have
3.9from which it is possible to deduce that
3.10and hence that
3.11where is defined as
3.12Thus, the Laplace–Beltrami operator in terms of the complex variables *ζ* and is
3.13with function as defined in (3.11).

We point out that the function
3.14appearing in (3.13) is the conformal factor, which gives
3.15where d*s* denotes an element of length on the surface of the torus.

## 4. Construction of solution

Let us write Green's function , to be determined, as the sum of two real-valued functions
4.1where is the solution to
4.2and is the solution to
4.3Here *ζ*_{0} corresponds to (*θ*_{0},*ϕ*_{0}) under the stereographic projection (3.4). Note that the existence of and satisfying (4.2) and (4.3) does not contradict Gauss's divergence theorem if we allow them both to be multi-valued.

### (a) Schottky–Klein prime function

We now introduce a special transcendental function known as the Schottky–Klein prime function associated with the annulus *D*_{ζ}. It can be defined through the infinite product
4.4This function has simple zeroes at *ζ*=*ρ*^{n}, , and satisfies the functional relation
4.5An accessible overview of the Schottky–Klein prime function is given by Crowdy [27]. We also introduce another special function that is related to *P*(*ζ*,*ρ*) by
4.6where *P*′(*ζ*,*ρ*) denotes the derivative of *P*(*ζ*,*ρ*) with respect to the first argument. This function has simple poles at *ζ*=*ρ*^{n}, (i.e. at the simple zeroes of *P*(*ζ*,*ρ*)), and satisfies the functional relation
4.7i.e. it is quasi-periodic as *ζ*↦*ρζ*.

### (b) The derivative

We first construct the derivative , as follows.

Let us first find . We propose
4.8for some . We can check that (4.8) satisfies (4.2) as follows. Let *T* denote the surface of the torus and let d*A* be an area element on *T*. Let d*σ* be an area element in the *ζ*-plane. We must have
4.9By transforming variables , it can be shown that
4.10By Green's theorem
4.11where ∂*D*_{ζ} denotes the boundary of *D*_{ζ}. Hence, we require
4.12Invoking the Argument Principle, and the fact that *K*(*ζ*/*ζ*_{0},*ρ*) has a single simple pole in *D*_{ζ} at *ζ*=*ζ*_{0} with residue 1, it can be shown that
4.13This completes our check of (4.8).

We now solve for . It follows immediately from (4.3) that
4.14It can be shown that
4.15where *η* is as in (3.12), *ς*_{1}(*ζ*) is an arbitrary function of *ζ* and *c* is the constant given by
4.16*c* and 1/*c* are the roots of *η*^{2}+2*αη*+1=0. Note that because *R*>*r*, it is evident that and *c*<−1. We have thus found that
4.17By adding (4.8) and (4.17), we see that the derivative of Green's function with respect to *ζ* is thus
4.18

### (c) Green's function

We now turn our attention to finding an explicit expression for Green's function using the result (4.18) of the previous section.

Integrating (4.8) with respect to *ζ* yields
4.19Here, we have chosen the arbitrary function of of integration in order that is indeed a purely real-valued function.

Next, integrating (4.17) with respect to *ζ*:
4.20The integral in (4.20) can be computed using (4.15) as follows. We shall make use of the dilogarithm function defined by:
4.21Li_{2}(*ζ*) is an analytic function of *ζ* except for branch points at 1 and . We choose a branch cut along the section of the real line in order to make it single-valued. A comprehensive discussion of the various properties of the dilogarithm function is given by Maximon [25]. Writing
4.22it is possible to show that
4.23Also,
4.24In (4.23) and (4.24), and are arbitrary functions of . Now, note the following property of Li_{2}(*ζ*), which can be deduced from its Taylor series expansion [25]:
4.25From (3.12), it may be deduced that |*η*|=1 for all *ζ*. Recall also that |*c*|>1 and so |*c*^{−1}*η*|<1 for all *ζ*. Hence, using (4.25) and the fact that , we have
4.26Furthermore, again using the facts that |*η*|=1 and , we have
4.27By combining the above, we arrive at
4.28where are constants (i.e. independent of both *ζ* and ), with
4.29*λ*_{4} and *λ*_{5} are yet to be specified owing to the arbitrary function of *ζ* in (4.15) and the arbitrary functions of in (4.23) and (4.24). We can determine *λ*_{4} and *λ*_{5} as follows. First recall that must be a real-valued function. Using this fact, it follows from (4.28) that *λ*_{4} and *λ*_{5} must both be real.

Furthermore, the complete Green's function must be single-valued on the torus and hence invariant with respect to the transformations *ζ*↦e^{2πi}*ζ* and *ζ*↦*ρζ*. It is straightforward to check that for any choice of *λ*_{4} and *λ*_{5}, the right-hand side of (4.28) is invariant with respect to *ζ*↦e^{2πi}*ζ*. Now consider *ζ*↦*ρζ*. It follows from (4.19) and (4.5) that
4.30provided we make the choice
4.31Next, it can be deduced from (3.12) that *ζ*↦*ρζ* corresponds to *η*↦e^{−2πi}*η*. Because |*η*|=1 for all *ζ*, *ζ*↦*ρζ* corresponds to *η*, moving once in the clockwise direction around the unit *η*-circle. Note that because |*c*|>1, then |*c*^{−1}*η*|<1, and so as *η*↦e^{−2πi}*η*, *c*^{−1}*η* does not go through the branch cut along associated with the dilogarithm function so the first term of (4.28) does not change. It follows from (4.28) that
4.32Hence, adding (4.30) and (4.32), it follows that we must have
4.33Finally, but may otherwise be chosen arbitrarily. We may wish to choose *λ*_{5} so that the complete Green's function satisfies a reciprocity condition.

Green's function for the operator is thus found to be
4.34with {*λ*_{j}|*j*=1,…,3} as in (4.29), *γ* as in (4.31), *c* as in (4.16), as in (3.12) and *λ*_{5}=0 (say).

We point out that it is straightforward to write 4.35It follows from well-known results of differential geometry that equals the Gaussian curvature of the toroidal surface. This gives a geometrical interpretation to one of the terms appearing in (4.34).

Finally, we mention that on differentiation of (4.34) with respect to *ζ*, the function *ς*_{1}(*ζ*) appearing in (4.15) may be identified as the constant
4.36

Figure 1 shows contour plots of in *D*_{ζ} for the torus . We chose three distinct values of the singularity *ζ*=*ζ*_{0}, and took them to be real and positive (without loss of generality). The location of the singularity *ζ*=*ζ*_{0} in each contour plot is apparent. For each value of *ζ*_{0}, we observe the existence of a critical point on the negative real axis, and two saddle points: one located on the positive real axis, and another located on the negative real axis close to the boundary. Inspection of the eigenvalues of the Hessian matrix of confirms the existence of the two saddle points, and reveals that the critical point is in fact a maximum. By our choice of sign of , *ζ*=*ζ*_{0} is a minimum.

Numerical experiments seem to suggest that under a change of the value of *α*, the number of saddle and critical points remains the same. However, changing *α* does have an effect on the qualitative appearance of the contours. More specifically, as , it appears that, for certain positions of the singularity *ζ*_{0}, there exist contours in the *ζ*-plane that encircle the inner boundary of *D*_{ζ}. These correspond to contours on the torus encircling its hole. As an example, illustrated in figure 2 are contours for the torus , with *ζ*_{0}=0.5. However, because *ρ* is very small for *α* close to 1 (for this particular case, *ρ*=0.00654), this behaviour is rather difficult to discern from the given plot. On the other hand, for *α*≫1, changing *α* does not appear to affect the appearance of the contours. Figure 3 shows contours for the torus , with *ζ*_{0}=0.7. One observes no qualitative difference in their appearance compared with those in figure 1. Similar behaviour is observed for larger *α*. Finally, we point out that our numerical experiments have been non-exhaustive, and a rigorous analysis of the behaviour of Green's function for the torus remains to be performed.

## 5. Application to vortex dynamics

In this section, we present an application of the foregoing theory to the field of vortex dynamics. We consider the flow of an infinitesimally thin layer of inviscid, incompressible fluid on . As will be shown, the Green's function we found in (4.34) is the streamfunction describing such a flow with a single point vortex surrounded by a uniform distribution of vorticity. The interested reader is referred to the monograph of Saffman [28] for an explanation of the key concepts used below.

The velocity field of the fluid **u** on is purely tangential to the surface; that is, in terms of the coordinates (*r*,*θ*,*ϕ*), we have
5.1for some *u*_{θ}, *u*_{ϕ}. Owing to the incompressibility condition , we may introduce the streamfunction *ψ*(*θ*,*ϕ*) such that
5.2It follows from (3.7) and (5.2) that, in terms of *ζ*, the velocity field can be expressed in the form
5.3Furthermore, it can be shown that the ‘image’ flow in the *ζ*-plane is given by
5.4where *U* and *V* denote, respectively, the components of the velocity in the real and imaginary directions.

Now introduce the scalar vorticity field *ω*(*θ*,*ϕ*) defined by
5.5Taking the curl of (5.2) and equating with (5.5) yields
5.6Because is a closed compact surface, it follows from Gauss's Divergence Theorem that we have an intrinsic topological constraint to enforce on **u**; that is,
5.7Consider now a point vortex on . This provides a *δ*-function distribution of vorticity. As a consequence of (5.7), a single point vortex cannot exist on unless an additional source of vorticity is present. One way to resolve this is as follows. Suppose the point vortex has circulation −1. By endowing the torus with a background sea of uniform vorticity *ω*_{0}=1/4*π*^{2}*rR*, the circulation associated with the point vortex is nullified; thus, the net circulation on will automatically be zero and the condition (5.7) satisfied. The streamfunction *ψ*(*θ*,*ϕ*) for such a system will now be deduced.

Suppose the point vortex is at (*θ*_{0},*ϕ*_{0}) on the surface of . Then *ψ*(*θ*,*ϕ*) satisfies the Laplace–Beltrami equation (2.3), in which physically, the Dirac delta function *δ*(*θ*,*ϕ*;*θ*_{0},*ϕ*_{0}) corresponds to the point vortex at (*θ*_{0},*ϕ*_{0}) while the additive constant term corresponds to the sea of uniform vorticity *ω*_{0} covering the surface of the torus. We thus identify *ψ*(*θ*,*ϕ*) as Green's function of found in (4.34).

Consider now the velocity field associated with such a vorticity distribution. It follows from (5.3) and (4.18) that this is given by
5.8where *ζ*_{0} is the image of the point vortex in the *ζ*-plane, and functions *K*(*ζ*,*ρ*) and are as in (4.6) and (4.15), respectively, recalling also that *ς*_{1}(*ζ*) is given by (4.36). Both *K*(*ζ*,*ρ*) and are quasi-periodic as *ζ*↦*ρζ*. We remark that there cannot exist doubly periodic functions in a period rectangle (and hence the equivalent concentric annulus) corresponding to the contributions to the velocity field of either the point vortex or the uniform sea of vorticity. We would thus expect a total velocity field to be the sum of two quasi-periodic functions: one corresponding to the point vortex, and one corresponding to the sea of uniform vorticity.

On physical grounds, however, the velocity field (5.8) must be single-valued on and hence invariant as *ζ*↦*ρζ* and *ζ*↦e^{2πi}*ζ*, or equivalently, as *η*↦e^{−2πi}*η* and *η*↦*η*, respectively. We now demonstrate this explicitly. First, note that the function is invariant with respect to both transformations. It is straightforward to check that (5.8) is invariant with respect to *ζ*↦e^{2πi}*ζ*. Next, consider this expression subject to *ζ*↦*ρζ*. Recall that and |*c*|>1. Then, from (4.15), it may be deduced that
5.9It thus follows from (4.7) and (5.9) that (5.8) is also invariant with respect to *ζ*↦*ρζ*, as required.

Finally, in the *ζ*-plane, we have from (5.4) that
5.10We remark that in (5.10), the simple pole of *K*(*ζ*/*ζ*_{0},*ρ*) at *ζ*=*ζ*_{0} gives rise to a singularity that is reminiscent of a planar point vortex. The pre-multiplying factor incorporates the curvature of the toroidal surface, and thus generalizes this singularity so that it pertains to a point vortex on a toroidal surface. Because the curvature of the toroidal surface is non-constant, it is not obvious whether the point vortex itself is advected by the velocity field (5.10). This issue is raised and discussed for a general closed surface in Boatto & Koiller [10]. However, clarification of this issue for the particular case of the torus remains a matter for future investigation.

## 6. Discussion

In this paper, we have derived, using techniques of complex analysis, an expression for Green's function of the Laplace–Beltrami operator on a toroidal surface. By using a stereographic projection to a planar concentric annulus, and performing the analysis in this projected plane, our Green's function admits a particularly concise form in terms of two special functions—the Schottky–Klein prime function and the dilogarithm function—yet still fully appreciates the topology of the torus.

The Schottky–Klein prime function is a natural candidate when considering problems in multiply connected circular domains. Having opted to construct Green's function in a complex plane, it was therefore natural to stereographically project the torus onto a concentric annulus and readily exploit the special properties of the Schottky–Klein prime function and its associated function theory in this circular domain. Furthermore, the related theory of Schottky groups naturally extends to provide uniformizations of more general higher genus surfaces. It may be possible to construct Green's functions for other surfaces in these terms by generalizing the analysis presented here.

It may be interesting to consider our formulation for Green's function in the limiting case as the major radius *R* becomes infinite with the minor radius *r* remaining fixed. Then, locally, the toroidal surface appears to be that of an infinitely long cylinder. In this limit, and the inner boundary of the annulus *D*_{ζ} approaches the outer. It may be more helpful to consider the analysis in this case in the rectangle in the conformally equivalent *Z*-plane, re-scaling by a factor of *R* so that this rectangle tends to an infinite strip of finite thickness. We may then use the Weierstrass elliptic functions that are related to *P*(*ζ*,*ρ*) and *K*(*ζ*,*ρ*) in a straightforward manner.

Green's function we have constructed should have a range of physical interpretations when considered as a solution to other problems of physical interest where the Laplace–Beltrami equation on a toroidal surface governs the system. For example, we have shown that the streamfunction for a point vortex on a toroidal surface is precisely Green's function constructed. Our Green's function should prove to be a convenient starting point for investigating these problems.

Keeping with the application to vortex dynamics, it would be interesting to emulate the various systems on a sphere considered by various authors [2,3,7,8,15–17] but on a toroidal surface. It would be of interest to analyse the possible similarities and differences of the exhibiting behaviours of the vortical systems between compact surfaces of differing genus. Such lines of enquiry are matters for future investigation.

## Acknowledgements

The authors thank D. G. Crowdy for helpful discussions and for suggesting this problem as a topic of study. C.C.G. acknowledges the support of an EPSRC studentship.

- Received August 13, 2012.
- Accepted October 1, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.