Banff International Research Station

13 January 2015

Green’s function for theLaplace-Beltrami operator

on a toroidal surface

Christopher GreenLindemann Trust Fellow

University of California San Diego

ccgreen@ucsd.edu

This is joint work with Jonathan Marshall

. – p.1

Introductory remarks

Green’s functions are fundamental objects and are very important in

mathematics (potential theory, complex analysis, numerical analysis

etc) and physics (fluid mechanics, electrostatics etc).

The simplest case is that of the Euclidean plane, and for this,

Green’s function is of course elementary:

∇2G = −δ(z − z0) ⇒ G(z, z0) = −1

2πlog |z − z0|.

Green’s function is also well-known for the case of the unit sphere:

∇2G = δ(θ − θ0, φ − φ0) −1

4π⇒

G(θ, φ; θ0, φ0) =1

4πlog(1−cos θ cos θ0−sin θ sin θ0 cos(φ−φ0)).

. – p.2

Introductory remarks

However, for more ‘complicated’ surfaces, the theory is

mathematically more complicated and far fewer explicit results are

known.

Ring torus is natural first choice on which to attempt to extend the

existing theory (being the simplest of these ‘complicated’ surfaces).

Herein, we will advocate an approach based on stereographicprojection combined with conformal mapping theory, and the judicioususe of certain special functions.

We will construct Green’s function for the ring toroidal surface in termsof a single planar complex variable.

. – p.3

Green’s function for the Laplace-Beltrami

operator on a spherical surface

For a unit sphere S, the Laplace-Beltrami operator ∇2S looks like

∇2S = (1 + ζζ)2

∂2

∂ζ∂ζ=

ζζ

F 2S(ζ, ζ)

∂2

∂ζ∂ζ, FS :=

|ζ|

(1 + ζζ).

where ζ = cot(θ/2) exp(iφ) is the stereographic projection of theunit sphere to the complex ζ-plane.

Green’s function GS(ζ, ζ) looks like

GS(ζ, ζ) =1

4πlog

[(ζ − ζ0)(ζ − ζ0)

(1 − ζζ)(1 − ζ0ζ0)

]

,

=1

2πlog |ω(ζ, ζ0)| + GS(ζ, ζ).

Here, ω(ζ, ζ0) = ζ − ζ0 is the Schottky-Klein prime functionassociated with the unit ζ-disc, and GS is an explicitly-known function.

D. G. Crowdy & M. Cloke, Analytical solutions for distributed multipolar vortex equilibriaon a sphere, Phys. Fluids (2003). . – p.4

Green’s function for the Laplace-Beltrami

operator on a tor oidal surface

First explicit representation of Green’s function GT (ζ, ζ) for

the Laplace-Beltrami operator ∇2T on a ring toroidal surface T , in

terms of a single complex variable ζ in a concentric annulus.

We found: ζ = Q(θ) exp(iφ),

∇2T ≡

ζζ

F 2T (ζ, ζ)

∂2

∂ζ∂ζ,

GT (ζ, ζ) =1

2πlog |ω(ζ, ζ0)| + GT (ζ, ζ).

Here, ω(ζ, ζ0) = P (ζ/ζ0) is the Schottky-Klein prime functionassociated with a concentric annulus in the ζ-plane, and P , FT andGT are explicitly-known functions. Note:

(1) similarity in the form of the Laplace-Beltrami operators ∇2S and ∇2

T ;

(2) similarity in the structure of Green’s functions GS and GT ;

(3) appearance of the Schottky-Klein prime function ω(ζ, ζ0).. – p.5

Motivation: vortical flows on the sphere

Considerable body of work on vortical motion on spherical surfaces

(largely motivated by the desire to study phenomena on Earth).

Reproduced from: [L] D. G. Crowdy, Point vortex motion on the surface of a sphere withimpenetrable boundaries, Phys. Fluids (2006); [R] D. G. Crowdy & M. Cloke, Analyticalsolutions for distributed multipolar vortex equilibria on a sphere, Phys. Fluids (2003).

Works include: Bogomolov (1977), Hally (1980), Kimura & Okamoto(1987), Kidambi & Newton (1998, 2000), Boatto & Koiller (2008), etc.

Kim (2010) analysed some point vortex motions on the spheroid.Kimura (1999) considered point vortex motion on the hyperbolic plane(via a Hamiltonian formalism).

For the future? Extend these studies to the torus (and other compactRiemann surfaces of higher genus). . – p.6

Motivation: quantized vortices and flows of

superfluid films on porus media

Exciting new applications! Behaviour of superfluids and quantized

vortices on porous media.

“Understanding the dynamics of quantized vortices bears tremendous importance...inspite of much progress made in the last decade, it should be pointed out that therigorous mathematical study of a large part of this subject on vortex dynamics remainsnearly non-existent" (Du, 2003, Contemp. Math.).

. – p.7

Ring torus Tr,R

Let Tr,R denote the ring torus of minor radius r and major radius R :

The surface of Tr,R consists of points (z, y, z) ∈ R3 where

x = (R − r cos θ) cos φ, y = (R − r cos θ) sin φ, z = r sin θ.

Pappus’ centroid theorem states that the surface area of a surface ofrevolution generated by rotating a plane curve about an axis externalto the curve and on the same plane is equal to the product of the arclength of the curve and the distance travelled by its geometric centroid.

So, the surface area of ring torus Tr,R is

A = (2πr)(2πR) = 4π2Rr.

Curvature of the toroidal surface is non-constant (Gaussian and meancurvatures determined from det and tr of the shape operator).

. – p.8

Key concept: double periodicity

The torus Tr,R is an intrinsically doubly-periodic object.

In terms of our complex variable ζ, Green’s function G(ζ, ζ) we shallconstruct must be doubly-periodic.

Quick aside: Also the interpretation of a torus as a rectangle (‘flat tori’).Solutions for these ‘flat tori’ problems are commonly constructed interms of elliptic functions in order to capture their requireddouble-periodicity.

. – p.9

Formulation of problem: the Laplace-Beltrami

operator ∇2Tr,R

on the surface of Tr,R

A straightforward exercise reveals that the Laplace-Beltrami operator

on the surface of Tr,R is

∇2Tr,R

=1

r2(R − r cos θ)

∂

∂θ

[

(R − r cos θ)∂

∂θ

]

+1

(R − r cos θ)2∂2

∂φ2.

We seek Green’s function G for this operator ∇2Tr,R

: this is areal-valued function and satisfies the partial differential equation

∇2Tr,R

G = δ(θ, φ; θ0, φ0) −1

4π2rR.

Note the constant term on the right-hand-side which is required by theGauss divergence theorem.

Goal: Solve for an explicit representation of Green’s function G bytransforming the problem to the complex ζ-plane.

Green’s function G(ζ, ζ) will need to be suitably doubly-periodic in ζ.. – p.10

Stereographic projection

Central to our construction of Green’s function G is the special

stereographic projection taking the surface of the torus Tr,R to a

concentric annulus in a complex ζ-plane.

To construct this stereographic projection, consider the composition oftwo separate mappings.

First, stereographically project Tr,R to a planar rectangle.

Akhiezer (1990) shows that the torus Tr,R can be stereographicallyprojected to a rectangle R of dimensions 2π × L in the Z-planethrough

Z = φ + i∫ θ

0

dθ′

α − cos θ′where

L =∫ 2 π

0

dθ′

α − cos θ′=

2π√

α2 − 1, and α =

R

r.

. – p.11

Stereographic projection

It is well-known that the conformal map from this rectangle R to a

concentric annulus D ζ = {ρ < |ζ| < 1} with ρ = e−L is just

ζ = exp (iZ) .

Thus, the stereographic projection of the surface of the torus Tr,R onto

the concentric annulus D ζ in a complex ζ-plane is

ζ = Q(θ) exp(iφ) = exp

[

−∫ θ

0

dθ′

α − cos θ′

]

exp (iφ) ,

or, upon evaluation of the integral,

ζ = exp

[

−2

√α2 − 1

tan−1

[√α + 1

α − 1tan(θ/2)

]]

exp (iφ) .

This stereographic projection is indeed conformal and one-to-one.. – p.12

Stereographic projection

ζ : Tr,R → D ζ

ζ = exp

[

−2

√α2 − 1

tan−1

[√α + 1

α − 1tan(θ/2)

]]

exp (iφ)

Exactly analogous in functional form to the stereographic projection tothe plane of the sphere, i.e. both are of the form

ζ = Q(θ) exp(iφ).

Note:

(1) ζ 7→ e2 π iζ corresponds to a rotation through 2π in the φ-direction;

(2) ζ 7→ ρζ corresponds to a rotation through 2π in the θ-direction.. – p.13

Stereographic projection

Using the facts that

∂

∂θ≡ −

1

α − cos θ

[

ζ∂

∂ζ+ ζ

∂

∂ζ

]

,∂

∂φ≡ i

[

ζ∂

∂ζ− ζ

∂

∂ζ

]

,

we can express the Laplace-Beltrami operator ∇2 in the following

nice form:∇2 =

4|ζ|2

(R − r cos θ)2∂2

∂ζ∂ζ.

It can be shown after some algebra that

F (ζ, ζ) := R − r cos θ ≡2rη/A2

η2 + 2αη + 1

so that ∇2Tr,R

can be expressed purely in terms of ζ and ζ:

∇2 =4|ζ|2

F 2(ζ, ζ)

∂2

∂ζ∂ζ,

where we have introduced - for convenience - the variable

η = |ζ|i/A, A =1

√α2 − 1

.. – p.14

Conformal factor

∇2 =4|ζ|2

F 2(ζ, ζ)

∂2

∂ζ∂ζ

Note that the function appearing in the above

h(ζ, ζ) =F (ζ, ζ)

|ζ|

is the conformal factor, which gives

ds2 = h2(ζ, ζ)(dq2 + q2dφ2

)

where ds denotes an element of length on the toroidal surface.

. – p.15

Construction of G(ζ, ζ0) for ∇2Tr,R

We construct Green’s function G(ζ, ζ) as the solution to the PDE

∇2Tr,R

G(ζ, ζ) = δ(ζ − ζ0) −1

4π2rR

by solving separately for the real-valued function G1(ζ, ζ) solving

∇2Tr,R

G1(ζ, ζ) ≡4|ζ|2

F 2(ζ, ζ)

∂2G1

∂ζ∂ζ= δ(ζ − ζ0),

and the real-valued function G2(ζ, ζ) solving

∇2Tr,R

G2(ζ, ζ) ≡4|ζ|2

F 2(ζ, ζ)

∂2G2

∂ζ∂ζ= −

1

4π2rR.

ThenG(ζ, ζ) = G1(ζ, ζ) + G2(ζ, ζ)

is the Green’s function we seek.

We find an explicit representation for this function G in terms of twospecial functions which we’ll now introduce.

. – p.16

Special function 1: P (ζ; ρ)

For the concentric annulus ρ < |ζ| < 1, introduce the special

transcendental function

P (ζ; ρ) = (1 − ζ)∞∏

k=1

(1 − ρkζ)(1 − ρkζ−1).

This is the Schottky-Klein prime function of the concentric annulus

(up to a multiplicative constant).

It satisfies the functional relation:

P (ρζ; ρ) = −ζ−1P (ζ; ρ).

Can define the following subsidiary function:

K(ζ; ρ) = ζP ′(ζ; ρ)

P (ζ; ρ).

This in turn satisfies the functional relation:

K(ρζ; ρ) = K(ζ; ρ) − 1,

i.e. it is quasi-periodic as ζ 7→ ρζ .. – p.17

The Schottky-Klein prime function

One form of the Schottky-Klein prime function is presented in

Baker’s 1897 monograph in the form of an infinite product:

ω(ζ, γ) = (ζ − γ)∏

θ ∈Θ′′

(ζ − θ(γ))(γ − θ(ζ))

(ζ − θ(ζ))(γ − θ(γ)).

Θ′′

⊂ Θ is a particular subset of the Schottky group Θ associated witha bounded M + 1 connected planar circular domain D ζ .

H. F. Baker, Abelian functions: Abel’s theorem and the allied theory of theta functions,Cambridge University Press (1897).

Key features: ω(ζ, γ) is:

(1) defined with respect to Θ;

(2) unique;

(3) analytic everywhere outside the 2M Schottky circles of Θ;

(4) ∼ (ζ − γ) as ζ → γ;

(5) has a known transformation property as ζ → θj(ζ).. – p.18

Concentric annulus as a planar model of a

ring torus (Schottky double)

Reproduced from: D. Mumford, C. Series & D. Wright, Indra’s Pearls: The Vision of FelixKlein, Cambridge University Press (2002).

The Schottky-Klein prime function/form is of great importance whenconstructing meromorphic functions with prescribed zeroes and poleson general Riemann surfaces.

. – p.19

Special function 2: Li2(ζ)

The second special function we will employ is the so-called

dilogarithm function Li2(ζ) which is defined by

Li2(ζ) = −∫ ζ

0

log(1 − u)

udu = −

∫ 1

0

log(1 − uζ)

udu.

Using either of these integral representations, one may expand thelogarithm in powers of ζ, obtaining the Taylor series expansion for thedilogarithm:

Li2(ζ) =∞∑

k=1

ζk

k2,

which is defined for all |ζ| ≤ 1.

The principal branch of the dilogarithm is defined by the integralsabove as a single-valued analytic function in the entire ζ-plane, withthe exception of branch points at 1 and ∞ (we choose the branch cutto be along the real line from 1 to ∞).

The integrals may be used to obtain analytic continuations of thedilogarithm for arguments outside the unit circle. . – p.20

Special function 2: Li2(ζ)

A comprehensive discussion of the various properties of the

dilogarithm function is given in the nice paper by Maximon (2003).

There are several remarkable identities involving the

dilogarithm function, several of which involve the golden ratio, see

e.g. Ramanujan (1919).

Lewin (1991) gives 67 dilogarithm identities (“ladders").

Useful property which can be deduced from the Taylor expansion:

Li2(ζ) = Li2(ζ), |ζ| < 1.

which in turn can be used to show that

Li2(c−1η) + Li2(c

−1η−1) = Re[Li2(c−1η)],

where we recall that η = |ζ|i/A, and here c = −α − 1/A.. – p.21

∂G1/∂ζ

Propose: ∂G1

∂ζ=

K(ζ/ζ0; ρ)

4πζ+

γ

2ζ.

Why is this? (1) ζ = Q(θ) exp(iφ) and ∇2Tr,R

are of the same

form as their spherical counterparts; this ultimately lead to

GS(ζ, ζ0) ∼1

2πlog |ζ − ζ0| ≡

1

2πlog |ω(ζ, ζ0)|, ζ → ζ0,

for ζ in a unit disc. Thus, we should expect

G(ζ, ζ0) ∼1

2πlog |P (ζ/ζ0; ρ)| ⇒

∂G

∂ζ∼

1

4π

K(ζ/ζ0; ρ)

ζ, ζ → ζ0.

for ζ in a concentric annulus.

(2) The additional term γ/2ζ comes about because, in the analyticextension of G(ζ, ζ0), one should expect a term of the form log ζ.(log ζ is the only non-constant analytic function in a concentric annulushaving constant values on the two boundary circles).

. – p.22

∂G1/∂ζ

(3) Need to check that∂G1

∂ζ=

K(ζ/ζ0; ρ)

4πζ+

γ

2ζsatisfies

∇2Tr,R

G1 = δ(ζ − ζ0).

Done by verifying that∫∫

T

∇2Tr,R

G1(θ, φ)dA =∫∫

T

δ(θ, φ; θ0, φ0)dA = 1

by transforming variables (θ, φ) → (ζ, ζ). Can be shown that

∫∫

T

∇2Tr,R

G1dA = 4∫∫

D ζ

∂2G1(ζ, ζ)

∂ζ∂ζdσ = −2i

∮

∂D ζ

∂G1(ζ, ζ)

∂ζdζ

and∮

∂D ζ

∂G1(ζ, ζ)

∂ζdζ =

∮

∂D ζ

[K(ζ/ζ0; ρ)

4πζ+

γ

2ζ

]

dζ =i

2

which completes the check. (4) Quasi-periodicity.... – p.23

∂G2/∂ζ

Find ∂G2/∂ζ by integrating directly:

∇2G2(ζ, ζ) ≡4ζζ

F 2(ζ, ζ)

∂2G2

∂ζ∂ζ= −

1

4π2rR

⇒ ξ(ζ, ζ) := ζ∂G2

∂ζ= −

1

16π2rR

∫ ζ F 2(ζ, ζ′)

ζ′dζ′.

After some algebra, it is found that

ξ(ζ, ζ) = −i

8π2

[

log

(η − c

η − c−2

)

+1

αA

(c

η − c+

c−1

η − c−1

)]

+ς1(ζ).

where η = |ζ|i/A and c = −α − 1/A. Thus, the derivative of Green’sfunction G(ζ, ζ) we seek with respect to ζ is

∂G

∂ζ=

K(ζ/ζ0; ρ)

4πζ+

γ

2ζ+

ξ(ζ, ζ)

ζ.

. – p.24

G1(ζ, ζ)

Integrating∂G1

∂ζ=

K(ζ/ζ0; ρ)

4πζ+

γ

2ζ

with respect to ζ yields

G1(ζ, ζ) =1

2πlog |P (ζ/ζ0; ρ)| + γ log |ζ|.

Here, we have chosen the arbitrary function of ζ of integration in orderthat G1(ζ, ζ) is indeed a purely real-valued function.

. – p.25

G2(ζ, ζ)

Next, we need to evaluate the integral

ξ(ζ, ζ) := ζ∂G2

∂ζ⇒ G2(ζ, ζ) =

∫ ζ ξ(ζ′, ζ)

ζ′dζ′.

After a lot of careful algebra...

G2(ζ, ζ) = λ1Re[Li2(c−1η)] + λ2 log |η − c| + λ3(log |ζ|)2

+λ4 log |ζ| + λ5

where the following are real constants:

λ1 =A

2π2, λ2 = −

1

2π2α, λ3 = −

1

8π2A.

Notice the dilogarithm function Li2(ζ) which played a crucial role indetermining this integral explicitly.

Recall Li2(ζ) is made single-valued by choosing a branch cut alongthe real axis from 1 to ∞. . – p.26

G(ζ, ζ0) needs to be doubl y-periodic

It is found that for G(ζ, ζ0) to be suitably real-valued, and invariant

as ζ 7→ e2 π iζ and ζ 7→ ρζ, we must take

λ4 = −1

4π

while λ5 is real and arbitrary (could be chosen so that G(ζ, ζ0)

satisfies a reciprocity condition). Further, we need

γ =log |ζ0|

4π2A.

Observation: we may express

log ||η| − c|2 = − log h(ζ, ζ) − log |ζ| + constant,

where recall h(ζ, ζ) = F (ζ, ζ)/|ζ| was the conformal factor. Fromdifferential geometry, −∇2

Tr,Rlog h(ζ, ζ) equals the Gaussian

curvature of the toroidal surface.. – p.27

Green’s function G(ζ, ζ0) is found

Green’s function G(ζ, ζ0) for the Laplace-Beltrami operator ∇2Tr,R

on the surface of Tr,R is thus found to be

G(ζ, ζ) =1

2πlog

∣∣∣∣P

(ζ

ζ0

; ρ

)∣∣∣∣+

A

2π2Re

[

Li2

(|ζ|i/A

c

)]

−1

2π2αlog

∣∣∣|ζ|i/A − c

∣∣∣ −

1

8π2A(log |ζ|)2+

(

γ −1

4π

)

log |ζ|

withγ =

log |ζ0|

4π2A, c = −α −

1

A, A =

1√

α2 − 1, α =

R

r.

It satisfies:∇2Tr,R

G(ζ, ζ) = δ(ζ − ζ0) −1

4π2rR.

It is invariant with respect to the transformations ζ 7→ e2 π iζ andζ 7→ ρζ i.e. it is suitably doubly-periodic (single-valued) everywhere onthe torus, as is necessary. . – p.28

Qualitative check: contours of G(ζ, ζ)

Contours of G(ζ, ζ) in the annulus 0.19744 < |ζ| < 1 for the torus

with r = 1 and R = 4:

Observe the existence of a critical point on the negative real axis, andtwo saddle points: one located on the positive real axis and anotherlocated on the negative real axis close to the boundary.

Inspection of the eigenvalues of the Hessian of G(ζ, ζ) confirms theexistence of this pair of saddle points and reveals that the critical pointis in fact a maximum (and ζ = ζ0 is a minimum).

. – p.29

Qualitative check: contours of G(ζ, ζ)

Our (non-exhaustive) numerical experiments ⇒ changing aspect ratioof the torus, the number of saddle and critical points remain the samebut does have an effect on the qualitative appearance of the contours.

Future work: rigorous analysis of the behaviour of Green’s functionG(ζ, ζ) for the torus! . – p.30

Application to vortex dynamics

Consider the flow of an infinitesimally thin layer of inviscid,

incompressible fluid on a torus.

Key fact: Green’s function G we have just found is the

streamfunction describing such a flow with a single point vortex

surrounded by a uniform distribution of vorticity.

Velocity field of the fluid u on the torus is purely tangential to thesurface:

u = (0, u θ , u φ ).

Owing to the incompressibility condition ∇ ∙ u = 0 , introducestreamfunction ψ(θ, φ) such that

u = ∇ψ(θ, φ) × (1, 0, 0) =

(

0,1

R − r cos θ

∂ψ

∂φ, −

1

r

∂ψ

∂θ

)

.

. – p.31

Application to vortex dynamics

u = ∇ψ(θ, φ) × (1, 0, 0) =

(

0,1

R − r cos θ

∂ψ

∂φ, −

1

r

∂ψ

∂θ

)

.

Velocity field u = (0, u θ , u φ ) can be expressed in the form

u φ − iu θ =2ζ

F (ζ, ζ)

∂ψ

∂ζ, F (ζ, ζ) =

2rη/A2

η2 + 2αη + 1.

Furthermore, it can be shown that the ‘image’ flow in the ζ-plane isgiven by

U − iV = −2i|ζ|2

F 2(ζ, ζ)

∂ψ

∂ζ.

. – p.32

Application to vortex dynamics

Consider a point vortex on the surface of the torus ⇒ a δ-function

distribution of vorticity.

The Gauss divergence theorem ⇒ a single point vortex can’t exist

on the torus unless an additional source of vorticity is present, so

that the net circulation on Tr,R is zero.

Resolution: If the point vortex has circulation −1, then endow the toruswith a background ‘sea’ of uniform vorticity

ω0 =1

4π2rR.

. – p.33

Green’s function as a streamfunction for a

vortical flow on Tr,R

Let’s now deduce the streamfunction ψ for this system.

Suppose the point vortex is at (θ0, φ0) on the surface of the torus.

Then ψ satisfies

∇2ψ = δ(θ, φ, θ0, φ0) −1

4π2rR.

Thus identify ψ as Green’s function of the Laplace-Beltrami operatoron the torus:

ψ ≡ G.

. – p.34

Velocity field is doubl y-periodic

Furthermore:

u φ −iu θ =2ζ

F (ζ, ζ)

∂G

∂ζ=

2ζ

F (ζ, ζ)

[1

4πK(ζ/ζ0; ρ) +

γ

2+ ξ(ζ, ζ)

]

.

This is invariant as ζ 7→ e2 π iζ and ζ 7→ ρζ, as is necessary.

And:

U−iV = −2i|ζ|2

F 2(ζ, ζ)

∂G

∂ζ= −

2i|ζ|2

F 2(ζ, ζ)

[K(ζ/ζ0; ρ)

4πζ+

γ

2ζ+

ξ(ζ, ζ)

ζ

]

.

Simple pole of K(ζ/ζ0; ρ) at ζ = ζ0 gives rise to a singularityreminiscent of a planar point vortex.Pre-multiplying factor incorporates the curvature of the toroidalsurface, and thus generalises the singularity so that it pertains to apoint vortex on a torus.Not obvious whether a single point vortex is itself advected by thevelocity field. . – p.35

Velocity field is doubl y-periodic

u φ −iu θ =2ζ

F (ζ, ζ)

∂G

∂ζ=

2ζ

F (ζ, ζ)

[1

4πK(ζ/ζ0; ρ) +

γ

2+ ξ(ζ, ζ)

]

.

General arguments in potential theory ⇒ the derivative ∂G/∂ζ to

be the sum of two quasi-periodic functions.

Recall that the two functions K(ζ; ρ) and ξ(ζ, ζ) are indeedquasi-periodic as ζ 7→ ρζ:

1

4πK(ρζ/ζ0; ρ) =

1

4π(K(ζ/ρ0; ρ) − 1) , ξ(ρζ, ρζ) = ξ(ζ, ζ)+

1

4π.

(easy to show invariance as ζ 7→ e2 π iζ).

Thus, the velocity field above can be made doubly-periodic.

. – p.36

Closing remarks

(1) The special geometrical nature of the ting toroidal surface the

has been fully incorporated into our modelling (via analytic

conformal projection).

(2) Our result will be able to be readily applied to solve other problemson a ring toroidal surface for which a type of Laplace-Beltrami equationturns out to be the governing equation.

(3) Shed light on interesting phenomena occurring in quantummechanics and flows of superfluid film vortices on toroidal and moregeneral holey surfaces.

(4) Evidence for developing the new theory for point vortex dynamicson holey surfaces by appealing to Schottky-Klein prime function / formmathematical framework. Tap into branches of geometric functiontheory such as exterior calculus and Hodge theory to make progress?(currently under investigation by CCG).

. – p.37

G(ζ, ζ) =1

2πlog

∣∣∣∣P

(ζ

ζ0

; ρ

)∣∣∣∣+

A

2π2Re

[

Li2

(|ζ|i/A

c

)]

−1

2π2αlog

∣∣∣|ζ|i/A − c

∣∣∣ −

1

8π2A(log |ζ|)2+

(

γ −1

4π

)

log |ζ|

Acknowledgements:

Helpful conversations with Björn Gustafsson (KTH, Sweden) and

Darren Crowdy (Imperial, UK).

The English-Speaking Union for a Lindemann Trust Fellowship.

Main reference:

C. C. Green & J. S. Marshall, Green’s function for the Laplace-Beltramioperator on a toroidal surface, Proc. R. Soc. A, 469, 2013.

. – p.38