3.2 Interaction Kernel
3.2.1 Existence, Uniqueness, and Properties of the Interaction Kernel
We saw from Lemma 3.2.1 that the logarithmic kernel distinguishes the point∞ ∈ Cˆ. Indeed, K(z,·) is a Bipolar Green’s function on ˆCwith poles atzand∞. In the sense of distributions we say that Khas a
”background measure” at∞. We are interested in symmetric kernels on compact Riemann surfaces with more general background measures. The reason is twofold. Firstly, because we do not want to unnecessarily provide preference to any point or measure, and more importantly, as we will later show in chapter four, there are interesting connections between the background measure, quadrature domains, and Laplacian growth.
Definition 3.2.2. The Interaction Kernel.
Let M be a compact Riemann surface with metric g compatible with its complex structure. Letνbe a finite positive Borel measure which we call the background measure. We define the Interaction Kernel, Gνto be a symmetric function in L2(M×M,volg⊗volg), satisfying
Z
M
Gν(z,w)∆gf(w)dvolg(w)=ν(M)f(z)− Z
M
f dν, (3.5)
for all f ∈C∞(M).
Lemma 3.2.3. The interaction kernel with background measureνis unique up to an additive constant.
The choice of the additive constant will be unimportant in what follows. By an abuse of terminology, for a fixed background measureν, we refer to a function satisfying Definition 3.2.2 astheinteraction kernel with background measureν.
Proof. Fixz ∈ M. LetG be another interaction kernel with background measure ν. Let{ψi}i∈N be an eigenbasis of the∆g, with corresponding eigenvalues{λi}in increasing order. Recall thatφ0is constant and thusφ0= q
1
|M|g. SinceGνandGare inL2(M×M,volg⊗volg), by hypothesis,
(Gν−G)(z,w)=X
i,j
ci,jφi(z)φj(w),
whereci,j∈C. Fori≥1, letf =λ1iφi, then
ci,j=Z
M×M
(Gν−G)(z,w)φi(w)φjdvolg(w)dvolg(z)
=Z
M
[ Z
M
(Gν−G)(z,w)∆gf dvolg(w)]φjdvolg(z)=0,
where the third equality follows from the fact thatGandGνboth satisfy Equation 3.5 and thus the expression in the inner bracket is zero. We thus haveci,j=0 fori>0. By the assumed symmetry ofGandGν, it follows thatci,j=cj,iand thusci,j=0 ifi>0 or j>0. Sinceφ0= q
1
|M|g, we have thus shown that (Gν−G)= c0,0
|M|g
.
We have thus shown that the interaction kernel is unique up to an additive constant.
Before considering examples, we remark about the relationship between the interaction kernel and Bipolar Green’s functions. LetGp,q(·) denote the Bipolar Green’s function with logarithmic pole at pand negative logarithmic pole atq. It is well-known thatGp,qsatisfies, the distributional identity,
Z
M
Gp,q(z)∆gf(z)dvolg(z)=2π(f(p)−f(q)) for anyf ∈C2(M). It follows from Fubini’s theorem that
Z
M
[ 1 2π
Z
M
Gp,q(z)dν(q)]∆gf(z)dvolg(z)=ν(M)f(p)− Z
M
f dν.
So 2π1 R
MGp,q(z)dν(q) satisfies the same distributional equation as Gν(p,·). The problem is thatGp,q(·) is uniquely determined up to an additive constant for each pair (p,q), and it is not immediately obvious how to choose a constant for each (p,q) such that 2π1 R
MGp,q(z)dν(q) is symmetric —that is 2π1 R
MGp,q(z)dν(q)=
1 2π
R
MGz,q(p)dν(q). We prefer to take a different approach to constructingGν. The following are some impor- tant examples of interaction kernels.
Example 3.2.4. M=Cˆ andν=2πδ∞. K is clearly symmetric, and by Lemma 3.2.1
Z
Cˆ
K(z,w)∆gf(w)dvolg(w)=2π(f(z)−f(∞))
=ν( ˆC)f(z)− Z
M
f dν,
so K is the interaction kernel onCˆ with background measure2πδ∞. Example 3.2.5. The Cylinder.
In this example we show that the potential on the cylinderC/Nobtained from the logarithmic potential by lifting toCcan be identified with the interaction kernel on the Riemann Sphere with background measure πδ0+πδ∞.
To determine the potential at a point z in the cylinderC/Nin the presence of a unit point charge at w∈C/N, we lift to the universal cover,C, of the cylinder. The preimage of the unit point charge at w under the covering map is{w˜ +n}n∈N, wherew is any lift of w. Let˜ ˜z be a lift of z. The potential atz in the presence of unit point˜ charges at{w˜ +n}n∈Nis formally the divergent series
X
n∈N
log 1
|˜z−w˜ +n|. (3.6)
However, we can regularize 3.6 by considering instead the limit
lim
N→∞
X
|n|<N
log 1
|˜z−w˜+n|− X
0<|n|<N
log 1
|n|−log(π).
Observe that
lim
N→∞
X
|n|<N
log 1
|z˜−w˜+n|− X
0<|n|<N
log 1
|n|−log(π)=− lim
N→∞log|π(˜z−w)| −˜ X
0<|n|<N
log|˜z−w˜ n +1|
=−log|π(˜z−w)˜
∞
Y
n=1
(1−(˜z−w)˜ 2 n )|
=−log|sin(π(˜z−w))|.˜
So the regularized potential islog|sin(π(˜1z−w))|˜ .This expression is independent of the choice of liftsw and˜ z and˜ so descends to a function on((C/N)×(C/N))\∆given by
log 1
|sin(π(z−w))|,
which is our desired potential on the cylinder. Letφ : C∗ → C/Nbe defined byφ(z) ≡ 2πi1 logz. φis a
conformal diffeomorphism so the potential onC∗is
log 1
|sin(π(φ(z)−φ(w)))| =log 2
|eπi(φ(z)−φ(w))−e−πi(φ(z)−φ(w))|
=log 2|eπiφ(z)+φ(w)|
|e2πiφ(z)−e2πiφ(w)|
=log2|√ zw|
|z−w|
=log 1
|z−w|−1 2log 1
|z|−1 2log 1
|w|−log 2.
Let G(z,w)≡log|z−w|1 −12log|z|1 −12log|w|1 −log 2. We claim that G is equal to the interaction kernel Gνon Cˆ with background measureν=πδ0+πδ∞. Since G is symmetric, we only need to check that it satisfies the distributional property. Let f ∈C∞( ˆC). Then
Z
Cˆ
G(z,w)∆gf(w)dvolg(w)
=Z
Cˆ
log 1
|z−w|∆gf(w)dvolg(w)−1 2
Z
Cˆ
log 1
|w|∆gf(w)dvolg(w)−1 2
Z
Cˆ
(log 1
|z| −log(2))∆gf(w)dvolg(w)
=2π(f(z)−f(∞))−π(f(0)−f(∞))−1 2(log 1
|z|−log(2)) Z
Cˆ
∆gf(w)dvolg(w)
=2πf(z)−πf(∞)−πf(0)
=ν( ˆC)f(z)− Z
Cˆ
f dν
where we used Lemma 3.2.1 twice in the second equality and the fact that Z
Cˆ
∆gf(w)dvolg(w)=0
in the third. We have thus shown that G=Gν.
Example 3.2.6. The torus with background measure induced by the flat metric.
In this example we compute the interaction kernel for the torus with background measure equal to the volume measure onT2induced by the flat metric, g. LetT2=C/ΛwhereΛ =h1, τiandτ∈H. Letνbe the volume measure induced by g. Note thatν(T2)==(τ). Letθ(z;τ)denote the Jacobi theta function. The Jacobi theta function satisfies the following identities:
θ(z+1, τ)=θ(z;τ), θ(z+τ;τ)=e−πiτe−2πizθ(z;τ).
We begin by defining a function G on(C×C)\∆and show that it is doubly periodic in each variable. Let G(z,w)= 1
2π[log 1
|θ(z−w)|+ 1
=(τ)(=(z−w))2].
Using the identities for the Jacobi theta function above, it is not difficult to verify that G is doubly periodic.
Therefore G descends to a functionG on˜ (T2×T2)\∆defined byG(π(z), π(w))˜ =G(z,w)whereπ:C→T2 is the canonical projection. Notice that G is symmetric and thus so isG. Also since G(z,˜ ·)has a logarithmic pole at z,G(π(z),˜ ·)has a logarithmic pole atπ(z). Moreover, offthe diagonal
∆G(z,·)= 1
2=(τ)∆(=(z− ·))2
= 2 2=τ
= 1
=τ.
Notice that off the diagonal Gν also satisfies∆gGν(z,·)volg = ν. It thus follows that Gν(z,·)−G(z,˜ ·) is harmonic and thus constant. By the symmetry of GνandG, it follows that G˜ ν=G˜+c where c is a constant.
Letgbe a Riemannian metric onM. The special case of interaction kernel corresponding toν= |M|1gvolghas been well studied and is called the Green’s function for the Laplace-Beltrami operator,∆g. Let{φi}∞i=0be a real orthonormal basis of eigenfunctions for∆gand let{λi}∞i=0be the corresponding eigenvalues in non-decreasing order. It is known that the kernel of∆gis precisely the constants and soφ0 = q
1
|M|g,λ0 =0, andλi >0 for i>0 since∆gis a nonnegative. Let
Gg(z,w)≡
∞
X
i=1
1
λiφi(z)φi(w).
By Weyl’s lemmaλiiasi→ ∞and so the series converges inL2(M×M).
The following properties ofGgare well-known i R
MGg(z,w)∆gψ(w)dvolg(w)=ψ(z)−|M|1
g
R
Mψdvolg, for anyψ∈C∞(M).
ii Gg(z,w) is smooth outside the diagonal∆⊂M×M.
iii Let (U, φ) be a coordinate chart thenGg(z,w)+2π1 log|φ(z)−φ(w)|is smooth onU×U.
iv For eachz∈M,||Gg(z,·)||L1(M,g)is uniformly bounded.
v R
MGg(z,w)dvolg=0 for eachz∈M.
vi Forz,w,∆gG(z,w)=−|M|1
g, where∆gacts on the first variable (or the second).
The Green’s function is an important example of an interaction kernel and we will show in Proposition 3.2.8 that it can be used to construct general interaction kernels. Before we can do this we need to prove two elementary lemmas.
Lemma 3.2.7. Let M and letνbe a finite positive Borel measure on M. Then for p>0
Gg ∈Lp(M×M, ν⊗volg).
Proof. SinceGg is smooth offthe diagonal, it suffices to show thatGg is in Lp in a neighborhood of the diagonal. SinceMis compact, there is covering
M=
n
[
i=1
Di,
whereDi=φ−1i (D) andφi:D→Diis a conformal map which extends conformally to a neighborhood ofD. Since
n
[
i=1
Di×Di⊃∆
is an open covering of the diagonal, it suffices to show thatGg ∈ Lp(Di×Di, ν⊗volg). Indeed, recall by property (iii),Gg(φ(z), φ(w))= 2π1 log|z−w|+ Ψ(z,w) whereΨis smooth. The result follows since
Z
Di×Di
|Gg(z,w)|pdν(z)dvolg(w)=Z
D×D
| 1
2πlog|z−w|+ Ψ(z,w)|pd(φ−1∗ ν)(z)d(φ−1∗ volg)(w)<∞, where the last inequality follows sinceφ−1∗ νis a finite andφ−1∗ volg is absolutely continuous with bounded
Radon-Nikodym derivative with respect to the area measure onC.
Lemma 3.2.8. Let M be a Riemann surface with metric g. Letνbe a background measure. Let f ∈C∞(M).
Then
Z
M
[ Z
M
Gg(w,s)dν(s)]∆gf(w)dvolg(w)=Z
M
f dν−ν(M)
|M|g
Z
M
f dvolg.
Proof. By Lemma 3.2.7,Gg ∈ L1(M×M,volg⊗ν). Moreover, since∆gf is bounded onM,Gg ∈ L1(M× M,∆gf volg⊗ν). By Fubini’s theorem the order of integration in the expression on the left side of the above equation can be interchanged. Using this and the distributional property (i) ofGgwe have
Z
M
[ Z
M
Gg(w,s)dν(s)]∆gf(w)dvolg(w)=Z
M
[ Z
M
Gg(w,s)∆gf(w)dvolg(w)]dν(s)
=Z
M
[f(s)− 1
|M|g
Z
M
f dvolg]dν(s)=Z
M
f dν−ν(M)
|M|g
Z
M
f dvolg.
The following proposition allows us to constructGνfromGg.
Proposition 3.2.9. Let M be a Riemann surface with metric g and letνbe a background measure. Then (up to an additive constant)
Gν(z,w)=ν(M)Gg(z,w)− Z
M
Gg(z,s)dν(s)− Z
M
Gg(w,s)dν(s). (3.7)
Moreover the right side of 3.7 has mean zero with respect to volg. Proof. Let
H(z,w)≡ν(M)Gg(z,w)− Z
M
Gg(z,s)dν(s)− Z
M
Gg(w,s)dν(s).
SinceGgis symmetric, so isH. By Lemma 3.2.7,H∈L2(M×M,volg⊗volg). All that remains is to show thatHsatisfies the required distributional property. Letf ∈C2(M), then
Z
M
H(z,w)∆gf(w)dvolg(w)
=ν(M)Z
M
Gg(z,w)∆gf(w)dvolg(w)− Z
M
[ Z
M
Gg(z,s)dν(s)]∆gf(w)dvolg(w)
− Z
M
[ Z
M
Gg(w,s)dν(s)]∆gf(w)dvolg(w)
=ν(M)(f(z)− 1
|M|g Z
M
f dvolg)−[ Z
M
Gg(z,s)dν(s)]Z
M
∆gf(w)dvolg(w)
− Z
M
f dν+ν(M)
|M|g Z
M
f dvolg
=ν(M)f(z)−ν(M)
|M|g
Z
M
f dvolg+0− Z
M
f dν+ν(M)
|M|g
Z
M
f dvolg
=ν(M)f(z)− Z
M
f dν.
We have used Lemma 3.2.8 in the second equality and thatR
M∆gf(w)dvolg(w)=0 in the third equality. We have thus shown thatH=Gν. Next we show thatR
MH(z,w)dvolg(w)=0. Indeed, Z
M
H(z,w)dvolg(w)=ν(M)Z
M
Gg(z,w)dvolg(w)− Z
M
[ Z
M
Gg(z,s)dν(s)]dvolg(z)
− Z
M
[ Z
M
Gg(w,s)dν(s)]dvolg(w)
=0− Z
M
[ Z
M
Gg(z,s)dvolg(z)]dν(s)− Z
M
[ Z
M
Gg(w,s)dvolg(w)]dν(s)=0,
where we have used Fubini’s theorem (which is justified in Lemma 3.2.7), and the fact that Z
M
Gg(z,w)dvolg(w)=0, which is property (v) ofGg.