# Dispersionless version of the constrained Toda hierarchy and symmetric
radial Löwner equation

T. Takebe  National Research University Higher School of Economics, 20
Myasnitskaya Ulitsa, Moscow 101000, Russian Federation, e-mail<EMAIL_ADDRESS>A. Zabrodin  Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina str. 8, Moscow, 119991, Russian Federation and Skolkovo Institute of
Science and Technology, 143026, Moscow, Russian Federation, e-mail:
<EMAIL_ADDRESS>
(April 2022)

We study the dispersionless version of the recently introduced constrained
Toda hierarchy. Like the Toda lattice itself, it admits three equivalent
formulations: the formulation in terms of Lax equations, the formulation of
the Zakharov-Shabat type and the formulation through the generating equation
for the dispersionless limit of logarithm of the tau-function. We show that
the dispersionless constrained Toda hierarchy describes conformal maps of
reflection-symmetric planar domains to the exterior of the unit disc. We also
find finite-dimensional reductions of the hierarchy and show that they are
characterized by a differential equation of the Löwner type which we call the
symmetric radial Löwner equation. It is also shown that solutions to the
symmetric radial Löwner equation are conformal maps of the exterior of the
unit circle with two symmetric slits to the exterior of the unit circle.

###### Contents

  1. 1 Introduction
  2. 2 Constrained Toda hierarchy
  3. 3 The dispersionless limit
  4. 4 Dispersionless constrained Toda hierarchy and conformal maps
  5. 5 One-variable reductions
  6. 6 Multivariable reductions
  7. 7 The symmetric radial Löwner equation
  8. 8 Conclusion
  9. A Appendix: Proof of the symmetric radial Löwner equation
    1. A.1 Behaviour of functions on the boundaries
    2. A.2 Symmetric radial Löwner equation for $\\{\omega_{s,t}(z)\\}$.
  10. Acknowledgements
  11. References

##  1 Introduction

The constrained Toda hierarchy was recently introduced in [2]. It is a
subhierarchy of the Toda lattice hierarchy [3] defined by the constraint

$\bar{L}=L^{{\dagger}}$ (1.1)

for the two pseudo-difference Lax operators $L$, $\bar{L}$ (in the symmetric
gauge). The constraint is preserved by the flows
$\partial_{t_{k}}-\partial_{t_{-k}}$ and is destroyed by the flows
$\partial_{t_{k}}+\partial_{t_{-k}}$ of the Toda lattice hierarchy, so one has
to fix $t_{k}+t_{-k}=0$. Here $\\{t_{k}\\}_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}}$ is the infinite set of independent variables (times) indexed by integer
numbers. The times $t_{k}$ with $k\neq 0$ are in general complex numbers while
$t_{0}=n$ is special: it takes integer values. The hierarchy is defined by the
infinite set of Lax equations (evolution equations for the Lax operator in the
times $t_{k}$).

An equivalent formulation is via the tau-function $\tau$ which is a function
of the infinite set of independent variables satisfying functional relations
which are bilinear in the case of the Toda lattice and have a more complicated
form for the constrained Toda hierarchy. In order to perform the
dispersionless limit [4, 5], one should substitute

$t_{k}\longrightarrow\frac{t_{k}}{\hbar},\qquad\tau(\\{t_{i}/\hbar\\})=e^{F(\\{t_{i}\\},\hbar)/\hbar^{2}}$
(1.2)

and take the $\hbar\to 0$ limit which allows one to obtain an equation for the
function $F=\lim\limits_{\hbar\to 0}F(\\{t_{i}\\},\hbar)$ from the equation
for the tau-function.

Around 2000 it was observed that the dispersionless hierarchies are deeply
related to the theory of univalent functions and in particular to conformal
maps of planar domains. This observation was developed in two seemingly
different but related directions. One of them treats equations of the
dispersionless Toda hierarchy as governing equations for conformal maps of
planar simply connected domains with smooth boundary as functions of their
harmonic moments which are identified with the hierarchical times $t_{k}$ [6,
7]. Another one is related to conformal maps of domains with curved slits. In
the seminal papers [8, 9] it was shown that one-variable reductions of the
dispersionless Kadomtsev-Petviashvili (KP) hierarchy are classified by
solutions of a Löwner-type differential equation which characterizes one-
parameter families of conformal mappings of domains with a growing slit onto a
fixed reference domain [10]–[13]. Later this important observation was
extended to hierarchies of other types and other versions of the Löwner
equation [14]–[20].

The aim of this paper is to suggest the dispersionless version of the
constrained Toda hierarchy (Section 3) and clarify its relation to conformal
maps (Section 4). We will show that equations of the dispersionless
constrained Toda hierarchy govern conformal maps of domains symmetric under
reflection with respect to the real axis (reflection-symmetric domains).

Furthermore, we study finite-dimensional reductions of the dispersionless
constrained Toda hierarchy (Sections 5,6). The meaning of reduction is as
follows. Infinite hierarchies of partial differential equations contain an
infinite number of independent variables (“times”) and an infinite number of
dependent variables. The simplest possible reduction (one-variable reduction)
is a reduction to just one dependent variable which depends on all the times.
All other dependent variables become functions of it. We will show that the
one-variable reductions of the dispersionless constrained Toda hierarchy are
described by solutions of a single ordinary differential equation of Löwner
type which we call symmetric Löwner equation. The equation contains an
arbitrary function (the “driving function”) which characterizes the reduction.
Geometrically, the driving function characterizes the shape of the slit and
the single dependent variable is a parameter along the slit. One can also
consider multivariable ($N$-variable) reductions when there are $N$ dependent
variables. In this case the reduction is described by a system of $N$
symmetric Löwner equations with $N$ driving functions.

Finally, Section 7 is devoted to clarifying the geometric meaning of the
symmetric radial Löwner equation. We show that solutions of this equation are
conformal maps of domains with two curved slits symmetric under reflection
with respect to the real axis. In the appendix we give a rigorous proof of the
symmetric radial Löwner equation.

##  2 Constrained Toda hierarchy

We begin with the Toda lattice hierarchy [3] in the symmetric gauge [21, 22].
The two Lax operators are

$L=c(n)e^{\partial_{n}}+\sum_{k\geq
0}U_{k}(n)e^{-k\partial_{n}},\quad\bar{L}=c(n-1)e^{-\partial_{n}}+\sum_{k\geq
0}\bar{U}_{k}(n)e^{k\partial_{n}},$ (2.1)

where $e^{k\partial_{n}}$ are shift operators acting on functions of $n$ as
$e^{k\partial_{n}}f(n)=f(n+k)$. The Lax operators are pseudo-difference
operators (infinite Laurent series in the shift operator). Given the Lax
operators, one can introduce the difference operators

$B_{m}=(L^{m})_{>0}+\frac{1}{2}(L^{m})_{0},\quad
B_{-m}=(\bar{L}^{m})_{<0}+\frac{1}{2}(\bar{L}^{m})_{0},\quad m=1,2,3,\ldots,$
(2.2)

where for a subset $\mathbb{S}\subset\mathbb{Z}$, we denote
$\displaystyle{\Bigl{(}\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}}U_{k}e^{k\partial_{n}}\Bigr{)}_{\raise-1.0pt\hbox{$\mbox{\Bbbb
S}$}}=\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb S}$}}U_{k}e^{k\partial_{n}}}$.
The Toda lattice hierarchy is given by the Lax equations

$\partial_{t_{m}}L=[B_{m},L],\quad\partial_{t_{m}}\bar{L}=[B_{m},\bar{L}]$
(2.3)

which define the hierarchical flows parametrized by the times $t_{m}$ for any
non-zero integer $m$. An equivalent formulation is through the zero curvature
(Zakharov-Shabat) equations

$\partial_{t_{k}}B_{m}-\partial_{t_{m}}B_{k}+[B_{m},B_{k}]=0.$ (2.4)

They encode differential-difference equations for the coefficient functions
$c(n)$, $U_{k}(n)$, $\bar{U}_{k}(n)$.

The constrained Toda hierarchy is obtained by imposing the constraint

$\bar{L}=L^{{\dagger}},$ (2.5)

where the †-operation is defined as $(f(n)\circ
e^{k\partial_{n}})^{{\dagger}}=e^{-k\partial_{n}}\circ f(n)$. This implies
that $\bar{U}_{k}(n)=U_{k}(n+k)$. It is easy to see that the constraint is
preserved by the flows $\partial_{t_{k}}-\partial_{t_{-k}}$ and is destroyed
by the flows $\partial_{t_{k}}+\partial_{t_{-k}}$, so one has to fix
$t_{k}+t_{-k}=0$. In this way all the coefficient functions can be regarded as
functions of $t_{k}$ with $k>0$ only. Introducing difference operators

$A_{m}=B_{m}-B_{-m},$ (2.6)

we can write the Lax and Zakharov-Shabat equations of the constrained
hierarchy in the form

$\partial_{t_{m}}L=[A_{m},L],\quad[\partial_{t_{m}}-A_{m},\,\partial_{t_{k}}-A_{k}]=0,\quad
m>0.$ (2.7)

The Zakharov-Shabat and Lax equations are compatibility conditions for the
linear problems

$\partial_{t_{m}}\psi=A_{m}\psi,\quad L\psi=z\psi$ (2.8)

for the wave function $\psi=\psi(n,{\bf t},z)$ depending on the spectral
parameter $z\in\mathbb{C}$ (and on all the times ${\bf
t}=\\{t_{1},t_{2},t_{3},\ldots\\}$). The set of linear equations is equivalent
to the bilinear relation [2]

$\Bigl{(}\oint_{C_{\infty}}-\oint_{C_{0}}\Bigr{)}\psi(n,{\bf
t},z)\psi(n^{\prime},{\bf t}^{\prime},z^{-1})\frac{dz}{2\pi iz}=0$ (2.9)

valid for all $n,n^{\prime},{\bf t},{\bf t}^{\prime}$, where $C_{\infty}$,
$C_{0}$ are small contours around $\infty$ and $0$ respectively.

It was proven in [2] that there exists a tau-function $\tau_{n}({\bf t})$ of
the constrained Toda hierarchy, and the wave function can be consistently
expressed through the tau-function as follows:

$\psi(n,{\bf t},z)=z^{n}e^{\xi({\bf t},z)}G_{n}^{1/2}({\bf
t},z)\frac{\tau_{n}({\bf t}-[z^{-1}])}{\tau_{n}({\bf t})},\quad z\to\infty,$
(2.10) $\psi(n,{\bf t},z^{-1})=z^{-n}e^{-\xi({\bf
t},z)}\tilde{G}_{n}^{1/2}({\bf t},z)\frac{\tau_{n+1}({\bf t})}{\tau_{n}({\bf
t})}\,\frac{\tau_{n+1}({\bf t}+[z^{-1}])}{\tau_{n}({\bf t})},\quad
z\to\infty.$ (2.11)

Here

$\xi({\bf t},z)=\sum_{k\geq 1}t_{k}z^{k},$ (2.12) $\begin{array}[]{c}{\bf
t}\pm[z^{-1}]=\Bigl{\\{}t_{1}\pm
z^{-1},t_{2}\pm\frac{1}{2}z^{-2},t_{3}\pm\frac{1}{3}z^{-3},\ldots\Bigr{\\}},\end{array}$
(2.13)

and

$G_{n}({\bf t},z)=1-z^{-2}\left(\frac{\tau_{n+1}({\bf t})\tau_{n-1}({\bf
t}-[z^{-1}])}{\tau_{n}({\bf t})\tau_{n}({\bf t}-[z^{-1}])}\right)^{2},$ (2.14)
$\tilde{G}_{n}({\bf t},z)=1-z^{-2}\left(\frac{\tau_{n}({\bf t})\tau_{n+2}({\bf
t}+[z^{-1}])}{\tau_{n+1}({\bf t})\tau_{n+1}({\bf t}+[z^{-1}])}\right)^{2}.$
(2.15)

Substituting these expressions into (2.9), one obtains an integral
equation111Unlike the Toda lattice hierarchy, this equation is not bilinear
but has a more complicated form with square roots of (2.14), (2.15). for the
tau-function which is the generating equation of the hierarchy. In particular,
setting ${\bf t}-{\bf t}^{\prime}=[a^{-1}]+[b^{-1}]$, $n-n^{\prime}=2$, the
residue calculus yields:

$\begin{array}[]{l}\displaystyle{\frac{a^{2}b}{a-b}G_{n}^{1/2}({\bf
t},a)\tilde{G}_{n-2}^{1/2}({\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}],a)\frac{\tau_{n}({\bf
t}\\!-\\![a^{-1}])}{\tau_{n}({\bf t})}\frac{\tau_{n-1}({\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])\tau_{n-1}({\bf
t}\\!-\\![b^{-1}])}{\tau^{2}_{n-2}({\bf t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])}}\\\
\\\ \displaystyle{-\frac{ab^{2}}{a\\!-\\!b}G_{n}^{1/2}({\bf
t},b)\tilde{G}_{n-2}^{1/2}({\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}],b)\frac{\tau_{n}({\bf
t}\\!-\\![b^{-1}])}{\tau_{n}({\bf t})}\frac{\tau_{n-1}({\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])\tau_{n-1}({\bf
t}\\!-\\![a^{-1}])}{\tau^{2}_{n-2}({\bf t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])}}\\\
\\\ \phantom{aaaaaaaaaaa}\displaystyle{=ab\frac{\tau^{2}_{n-1}({\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])}{\tau^{2}_{n-2},{\bf
t}\\!-\\![a^{-1}]\\!-\\![b^{-1}])}-(ab)^{-1}\frac{\tau^{2}_{n+1}({\bf
t})}{\tau_{n}^{2}({\bf t})}.}\end{array}$ (2.16)

Finally, let us point out the relation between the tau-function of the
constrained Toda hierarchy and the tau-function $\tau_{n}^{\rm Toda}(\ldots
t_{-2},t_{-1};t_{1},t_{2},\ldots)$ of the Toda lattice hierarchy satisfying
the constraints

$(\partial_{t_{k}}+\partial_{t_{-k}})\log\tau^{\rm
Toda}\Bigr{|}_{t_{j}+t_{-j}=0}=0\quad\mbox{for all $k,j$ and $t_{0}$,
$t_{j}-t_{-j}$}$ (2.17)

which are equivalent to (2.5). We have (see [2]):

$\tau_{n}({\bf t})=\sqrt{\vphantom{A^{\oint}}\tau_{n}^{\rm Toda}(\ldots-
t_{2},-t_{1};t_{1},t_{2},\ldots)}.$ (2.18)

##  3 The dispersionless limit

In the dispersionless limit [4, 5], one should substitute

$t_{k}\longrightarrow\frac{t_{k}}{\hbar},\qquad\tau_{t_{0}/\hbar}({\bf
t}/\hbar)=e^{F(t_{0},{\bf t},\hbar)/\hbar^{2}}$ (3.1)

and take the $\hbar\to 0$ limit which allows one to obtain an equation for the
function $F=\lim\limits_{\hbar\to 0}F(t_{0},{\bf t},\hbar)$ from the equation
for the tau-function. In accordance with this prescription, we can write

$\tau_{n\pm 1}({\bf
t})\longrightarrow\exp\Bigl{(}\hbar^{-2}e^{\pm\hbar\partial_{t_{0}}}F\Bigr{)},$
$\tau_{n}({\bf
t}\pm[a^{-1}])\longrightarrow\exp\Bigl{(}\hbar^{-2}F(t_{0},t\pm\hbar[a^{-1}],\hbar)\Bigr{)}=\exp\Bigl{(}\hbar^{-2}e^{\pm\hbar
D(a)}F\Bigr{)},$

where $D(z)$ is the differential operator

$D(z)=\sum_{k\geq 1}\frac{z^{-k}}{k}\,\partial_{t_{k}}.$ (3.2)

The dispersionless limit of equation (2.16) is then straightforward. We only
note that

$\lim_{\hbar\to 0}G_{t_{0}/\hbar}({\bf t}/\hbar,z)=\lim_{\hbar\to
0}\tilde{G}_{t_{0}/\hbar}({\bf
t}/\hbar,z)=1-z^{-2}e^{2\partial_{t_{0}}(\partial_{t_{0}}+D(z))F},$ (3.3)

so the square roots in (2.17) disappear in the $\hbar\to 0$ limit. The final
result is

$\frac{ae^{-\partial_{t_{0}}(\partial_{t_{0}}+D(a))F}\\!-\\!be^{-\partial_{t_{0}}(\partial_{t_{0}}+D(b))F}}{a-b}=\frac{abe^{-\partial_{t_{0}}(2\partial_{t_{0}}+D(a)+D(b))F}\\!-\\!1}{ab}\,e^{(\partial_{t_{0}}+D(a))(\partial_{t_{0}}+D(b))F}.$
(3.4)

Introducing the function

$w(z)=ze^{-\partial_{t_{0}}(\partial_{t_{0}}+D(z))F},$ (3.5)

we can write this equation in the form

$\frac{w(a)-w(b)}{w(a)w(b)-1}=(b^{-1}-a^{-1})e^{(\partial_{t_{0}}+D(a))(\partial_{t_{0}}+D(b))F}.$
(3.6)

Taking logarithm of both sides and applying the $t_{0}$-derivative, we can
represent it in the form of a closed equation for the function $w(z)$:

$(\partial_{t_{0}}+D(a))\log
w(b)=-\partial_{t_{0}}\log\frac{w(a)-w(b)}{w(a)w(b)-1}.$ (3.7)

It is instructive to rewrite this equation in terms of the inverse function to
$w(z)$ which we denote as $z(w)$. Noting the relation

$\partial_{t_{k}}w(z)=-\frac{\partial_{t_{k}}z(w)}{\partial_{w}z(w)}$

between the partial derivatives, and, as a consequence,

$w\partial_{w}z(w)\,D(a)\log w(z)=-D(a)z(w)$

(the derivative in the right hand side is taken at constant $w$), we can
rewrite equation (3.7) in the form

$D(a)z(w)=\left\\{z(w),\,\log\frac{w(a)-w}{w(a)-w^{-1}}\right\\},$ (3.8)

where

$\\{f,g\\}=w\frac{\partial f}{\partial w}\frac{\partial g}{\partial
t_{0}}-w\frac{\partial f}{\partial t_{0}}\frac{\partial g}{\partial w}$ (3.9)

is the Poisson bracket. Equation (3.8) is the generating Lax equation for the
dispersionless constrained Toda hierarchy. To see this, we introduce the Faber
polynomials $B_{k}(w)$ according to the expansion

$-\log\Bigl{(}w(z)-w\Bigr{)}+\frac{1}{2}\log\frac{zw(z)}{r}=\sum_{k\geq
1}\frac{z^{-k}}{k}\,B_{k}(w),\quad r=(\lim_{z\to\infty}(w(z)/z))^{-1}.$ (3.10)

Here $w$ is an independent variable ($w\neq w(z)$). Then equation (3.8)
implies the Lax equations of the dispersionless constrained Toda hierarchy:

$\partial_{t_{k}}z(w)=\Bigl{\\{}B_{k}(w)-B_{k}(w^{-1}),\,z(w)\Bigr{\\}}.$
(3.11)

It is easy to see that

$\begin{array}[]{c}B_{k}(w)=\Bigl{(}z^{k}(w)\Bigr{)}_{>0}+\frac{1}{2}\Bigl{(}z^{k}(w)\Bigr{)}_{0}.\end{array}$
(3.12)

Comparing with (2.1), (2.6), we see that the function (or rather the Laurent
series) $z(w)$ is the dispersionless limit of the Lax operator:
$e^{\partial_{n}}\to w$, $L\to z(w)$ and the formulas (3.11), (3.12) are
dispersionless limits of (2.7), (2.2).

From now on, keeping in mind applications to conformal maps of planar domains,
we we will consider the real form of the Toda lattice hierarchy such that the
time variables satisfy the conditions

$t_{-k}=-\bar{t}_{k},\quad k\geq 1,\quad\mbox{and $t_{0}$ is real}.$ (3.13)

Moreover, the tau-function of the real form of the Toda lattice hierarchy is
real. For the constrained Toda hierarchy this means that $t_{k}=\bar{t}_{k}$,
i.e., all the time variables are real.

The dispersionless limit of the Lax operator $L$ is the Laurent series $z(w)$
while the limit of the second Lax operator, $\bar{L}$, is
$\bar{z}(w^{-1})=\overline{z(\bar{w}^{-1})}$. Since $e^{\partial_{n}}$ in the
dispersionless limit becomes $w$ and
$(e^{\partial_{n}})^{{\dagger}}=e^{-\partial_{n}}$, we have
$w^{{\dagger}}=w^{-1}$, so the constraint (2.5) in the dispersionless limit
implies that $z(w)=\overline{z(\bar{w})}$, or $w(z)=\overline{w(\bar{z})}$,
i.e. all coefficients of the Laurent series $z(w)$ and $w(z)$ are real. The
constraints (2.17) for the tau-function then mean that
$v_{k}=\partial_{t_{k}}F$ are also real.

##  4 Dispersionless constrained Toda hierarchy and conformal maps

It is known that the real reduction of the dispersionless Toda lattice
hierarchy governs conformal maps of planar domains [7, 23, 24]. Let us recall
the main facts related to this connection.

Given a simply-connected compact planar domain ${\sf D}$ with a smooth
boundary, one can define the harmonic moments

$v_{k}=\frac{1}{\pi}\int_{{\sf D}}z^{k}d^{2}z,\quad k\geq 1,\quad d^{2}z\equiv
dxdy$ (4.1)

together with the complimentary moments of the exterior domain
$\mathbb{C}\smallsetminus{\sf D}$

$t_{k}=-\frac{1}{\pi k}\int_{\raise-1.0pt\hbox{$\mbox{\Bbbb
C}$}\smallsetminus{\sf D}}z^{-k}d^{2}z.$ (4.2)

Without loss of generality we assume that the domain ${\sf D}$ contains the
origin. The moments $t_{k}$ and $v_{k}$ are in general complex numbers. We
also denote

$t_{0}=\frac{1}{\pi}\int_{\sf D}d^{2}z=\frac{\mbox{Area of ${\sf D}$}}{\pi}$
(4.3)

which is a real number.

Let $z(w)$ be the conformal map from the exterior of the unit circle to the
domain $\mathbb{C}\smallsetminus{\sf D}$ normalized in such a way that
$z(\infty)=\infty$ and $z^{\prime}(\infty)=r>0$ ($r$ is a real positive number
called conformal radius). As it was shown in [7], the function $z(w)$
satisfies the equations

$\partial_{t_{k}}z(w)=\\{B_{k}(w),z(w)\\},\quad\partial_{\bar{t}_{k}}z(w)=-\\{\bar{B}_{k}(w^{-1}),z(w)\\}$
(4.4)

with $B_{k}(w)$ given by (3.12). This allows one to identify $z(w)$ with the
Lax function of the dispersionless Toda hierarchy (the dispersionless limit of
the Lax operator).

The dispersionless limit of logarithm of tau-function (the $F$-function) is
given by

$F^{\rm Toda}=-\frac{1}{\pi^{2}}\int_{\sf D}\int_{\sf
D}d^{2}zd^{2}\zeta\log\Bigl{|}z^{-1}-\zeta^{-1}\Bigr{|}$ (4.5)

(see [23]). It is a real-valued function of $t_{0}$, $t_{k}$ and
$\bar{t}_{k}$. The first derivatives of $F$ with respect to $t_{k}$’s yield
the complimentary moments $v_{k}$:

$v_{k}=\partial_{t_{k}}F^{\rm Toda}.$ (4.6)

The second mixed derivatives of $F$ provide full information about the
conformal map $w(z)$ from $\mathbb{C}\smallsetminus{\sf D}$ to the exterior of
the unit circle (the inverse function of $z(w)$) and the Green function
$G(z,\zeta)$ of the Dirichlet boundary problem in
$\mathbb{C}\smallsetminus{\sf D}$. Namely,

$\begin{array}[]{c}w(z)=z\exp\Bigl{(}-\frac{1}{2}\partial_{t_{0}}^{2}F^{\rm
Toda}-\partial_{t_{0}}D(z)F^{\rm Toda}\Bigr{)},\end{array}$ (4.7)
$G(z,\zeta)=\log\Bigl{|}z^{-1}-\zeta^{-1}\Bigr{|}+\frac{1}{2}\nabla(z)\nabla(\zeta)F^{\rm
Toda},$ (4.8)

where

$\nabla(z)=\partial_{t_{0}}+D(z)+\overline{D(z)}$

and $D(z)$ is the differential operator (3.2). The well-known formula

$G(z,\zeta)=\log\left|\frac{w(z)-w(\zeta)}{w(z)\overline{w(\zeta)}-1}\right|$
(4.9)

which expresses the Green function through the conformal map is equivalent to
the dispersionless limit of the Toda equations for the tau-function.

In the case of the constrained Toda hierarchy, the coefficients of the Laurent
series $z(w)$ and the moments $t_{k}$ are real. It is obvious from (4.2) that
for domains symmetric under reflection with respect to the real axis
(reflection-symmetric domains) all moments are real. The converse statement,
i.e. that for real moments the corresponding domain is symmetric, can be
proved by means of the Schwarz function222The Schwarz function $S(z)$ [25] is
a holomorphic function in a strip-like neighborhood of the boundary of the
domain such that $\bar{z}=S(z)$ on the boundary. It is a generating function
of the harmonic moments. Therefore, if all harmonic moments are real,
$\bar{z}=S(z)$ implies $z=S(\bar{z})$, which means that complex conjugation
preserves the boundary.. This means that we deal with conformal maps of
reflection-symmetric domains. The moments $v_{k}$ are also real, and formula
(4.6) means that the constraint (2.17) is satisfied if we are in the class of
reflection-symmetric domains. From (2.18) it follows that the $F$-function of
the dispersionless constrained Toda hierarchy is related to $F^{\rm Toda}$ as

$F^{\rm Toda}=2F.$ (4.10)

Therefore, one can see that formula (3.5) is a specialization of (4.7) for
reflection-symmetric domains. In a similar way, equation (3.6) is a
specialization of (4.8) for reflection-symmetric domains with $z,\zeta$ lying
on the real axis. In the class of reflection-symmetric domains, the function
$F$, regarded as a function of real moments $t_{k}$, satisfies equation (3.4).

As it follows from the result of [23], the function $F$ satisfies the
following quasi-homogeneity equation:

$2F=-\frac{1}{4}\,t_{0}^{2}+t_{0}\partial_{t_{0}}F+\sum_{k\geq
1}(2\\!-\\!k)t_{k}\partial_{t_{k}}F.$ (4.11)

Example. For the ellipse with half-axes $a,b$ centered at $x_{0}\in\mathbb{R}$
we have:

$t_{0}=ab,\quad t_{1}=\frac{2bx_{0}}{a+b},\quad 2t_{2}=\frac{a-b}{a+b},$

and all other moments are equal to zero. For the ellipse the function $F$ is

$F=\frac{1}{4}\,t_{0}^{2}\log
t_{0}-\frac{3}{8}\,t_{0}^{2}-\frac{1}{4}\,t_{0}^{2}\log(1-4t_{2}^{2})+\frac{t_{0}t_{1}^{2}}{2(1-2t_{2})}$
(4.12)

(see [7]).

##  5 One-variable reductions

In this section we consider reductions of the dispersionless constrained Toda
hierarchy. For one-variable reductions, the dependence of $w(z)$ on the times
is implemented by means of a single variable $\lambda=\lambda({\bf t})$, i.e.,
$w(z;{\bf t})=w(z,\lambda({\bf t}))$. This means that instead of the function
of infinitely many variables $w(z;{\bf t})$ we now deal with a function of two
variables $w(z,\lambda)$. Our goal is to find all possible forms of this
function consistent with the infinite hierarchy. The reduction is an
exceptional non-generic solution. We will see that such solutions correspond
to conformal maps of domains with slits.

The derivation below is parallel to that for reductions of the dispersionless
Toda hierarchy [20] (see also [17, 26]). Assuming that $w(z)=w(z,\lambda)$ and
using the chain rule of differentiating, we have:

$(\partial_{t_{0}}+D(\zeta))\log w(z)=\partial_{\lambda}\log
w(z)\cdot(\partial_{t_{0}}+D(\zeta))\lambda$

and

$(\partial_{t_{0}}+D(z))\lambda=\frac{(\partial_{t_{0}}+D(z))\log
r}{\partial_{\lambda}\log r}.$ (5.1)

We recall that $\log r=-\log\lim\limits_{z\to\infty}(w(z)/z)$. Tending
$b\to\infty$ in equation (3.7), we get:

$\partial_{t_{0}}\log w(z)=-(\partial_{t_{0}}+D(z))\log r.$ (5.2)

Therefore, equation (5.1) can be written as

$(\partial_{t_{0}}+D(z))\lambda=-\frac{\partial_{\lambda}\log
w(z)}{\partial_{\lambda}\log r}\,\partial_{t_{0}}\lambda.$ (5.3)

Substituting this into (3.7) and assuming that $\partial_{t_{0}}\lambda$ is
not identically zero, we get:

$\partial_{\lambda}\log w(z)\partial_{\lambda}\log
w(\zeta)=\partial_{\lambda}\log
r\left[\partial_{\lambda}w(z)\Bigl{(}\frac{1}{w(z)-w(\zeta)}-\frac{w(\zeta)}{w(z)w(\zeta)-1}\Bigr{)}\right.$
$\left.-\partial_{\lambda}w(\zeta)\Bigl{(}\frac{1}{w(z)-w(\zeta)}+\frac{w(z)}{w(z)w(\zeta)-1}\Bigr{)}\right]$

or, after rearranging,

$w(z)+w^{-1}(z)+\partial_{\lambda}\log
r\frac{w(z)-w^{-1}(z)}{\partial_{\lambda}\log
w(z)}=w(\zeta)+w^{-1}(\zeta)+\partial_{\lambda}\log
r\frac{w(\zeta)-w^{-1}(\zeta)}{\partial_{\lambda}\log w(\zeta)}.$

The left hand side depends only on $z$ while the right hand side depends only
on $\zeta$. It then follows from this equation that

$\eta(\lambda):=w(z)+w^{-1}(z)+\partial_{\lambda}\log
r\frac{w(z)-w^{-1}(z)}{\partial_{\lambda}\log w(z)}$ (5.4)

does not depend on $z$. Equation (5.4) can be read as the differential
equation for the function $w(z)=w(z,\lambda)$:

$\partial_{\lambda}\log
w(z)=-\frac{w(z)-w^{-1}(z)}{w(z)+w^{-1}(z)-\eta(\lambda)}\,\partial_{\lambda}\log
r,$ (5.5)

where $\eta(\lambda)$ is a real-valued function of $\lambda$. Setting

$\eta(\lambda)=e^{i\xi(\lambda)}+e^{-i\xi(\lambda)},$ (5.6)

where $\xi(\lambda)$ is another real-valued function of $\lambda$, we can
represent equation (5.5) in the form

$\partial_{\lambda}\log
w(z)=-\frac{1}{2}\left(\frac{w(z)+e^{i\xi(\lambda)}}{w(z)-e^{i\xi(\lambda)}}+\frac{w(z)+e^{-i\xi(\lambda)}}{w(z)-e^{-i\xi(\lambda)}}\right)\partial_{\lambda}\log
r.$ (5.7)

In particular, one can choose $\lambda=\log r$, then the equation simplifies:

$\frac{\partial\log w(z)}{\partial\log
r}=-\frac{1}{2}\left(\frac{w(z)+e^{i\xi}}{w(z)-e^{i\xi}}+\frac{w(z)+e^{-i\xi}}{w(z)-e^{-i\xi}}\right).$
(5.8)

We call equation (5.7) (or (5.8)) symmetric radial Löwner equation. Its right
hand side is the half-sum of the right hands sides of the radial Löwner
equations with the driving functions $\xi(\lambda)$ and $-\xi(\lambda)$. This
is similar to the quadrant Löwner equation [18] which appears as a reduction
condition in the dispersionless BKP (or CKP) hierarchy333As is shown in [27],
the dispersionless limits of the BKP and CKP hierarchies are the same. and
whose right hand side is the half-sum of the right hands sides of the chordal
Löwner equations with the driving functions $\xi(\lambda)$ and
$-\xi(\lambda)$. Moreover, in a certain scaling limit the symmetric radial
Löwner equation becomes the quadrant Löwner equation. To wit, setting
$w(z)=e^{i\epsilon p(z)}$, $r=e^{-\epsilon^{2}u/2}$,
$\xi(\lambda)\to\epsilon\xi(\lambda)$ and taking the limit $\epsilon\to 0$, we
obtain the quadrant Löwner equation

$\partial_{\lambda}p(z)=-\frac{1}{2}\left(\frac{1}{p(z)-\xi(\lambda)}+\frac{1}{p(z)+\xi(\lambda)}\right)\partial_{\lambda}u.$
(5.9)

As is shown in Section 7, solutions of the symmetric radial Löwner equation
are functions which define conformal mappings from the exterior of the unit
circle with two symmetric slits to the exterior of the unit circle.

The dependence of $\lambda$ on the times $t_{k}$ is determined by a system of
equations of the hydrodynamic type. They follow from equation (5.2) which can
be written as

$(\partial_{t_{0}}+D(z))\lambda=\frac{(\partial_{t_{0}}+D(z))\log
r}{\partial_{\lambda}\log r}=-\frac{\partial_{t_{0}}\log
w(z)}{\partial_{\lambda}\log r}=-\frac{\partial_{\lambda}\log
w(z)}{\partial_{\lambda}\log r}\,\partial_{t_{0}}\lambda.$

Plugging here the symmetric Löwner equation (5.7), we obtain

$D(z)\lambda=\left(\frac{e^{i\xi(\lambda)}}{w(z)-e^{i\xi(\lambda)}}+\frac{e^{-i\xi(\lambda)}}{w(z)-e^{-i\xi(\lambda)}}\right)\partial_{t_{0}}\lambda.$
(5.10)

Taking the $w$-derivative of equation (3.10) defining the Faber polynomials,
we get

$\frac{e^{\pm i\xi(\lambda)}}{w(z)-e^{\pm i\xi(\lambda)}}=\sum_{k\geq
1}\frac{z^{-k}}{k}\,\phi_{k}(e^{\pm i\xi}),\quad\phi_{k}(w)\equiv
w\partial_{w}B_{k}(w).$ (5.11)

Therefore, equation (5.10) is equivalent to the following infinite system of
equations of hydrodynamic type:

$\partial_{t_{k}}\lambda=\Bigl{(}\phi_{k}(e^{i\xi(\lambda)})+\phi_{k}(e^{-i\xi(\lambda)})\Bigr{)}\partial_{t_{0}}\lambda.$
(5.12)

The hodograph equation

$t_{0}+\sum_{k\geq
1}t_{k}\Bigl{(}\phi_{k}(e^{i\xi(\lambda)})+\phi_{k}(e^{-i\xi(\lambda)})\Bigr{)}=R(\lambda)$
(5.13)

gives a general solution in an implicit form. Here $R(\lambda)$ is an
arbitrary function of $\lambda$.

##  6 Multivariable reductions

A multivariable ($N$-variable) reduction of the dispersionless constrained
Toda hierarchy means that the function $w(z)$ depends on the times ${\bf t}$
through $N$ functions $\lambda_{j}=\lambda_{j}({\bf t})$, i.e., $w(z)=w(z;{\bf
t})=w(z;\lambda_{1}({\bf t}),\ldots,\lambda_{N}({\bf t}))$. Below we prove
that solutions of a system of symmetric radial Löwner equations give solutions
to the hierarchy.

We consider the system of $N$ symmetric radial Löwner equations of the form
(5.7) which characterize the dependence of
$w(z)=w(z;\lambda_{1},\ldots,\lambda_{N})$ on the variables $\lambda_{j}$:

$\frac{\partial\log
w(z)}{\partial\lambda_{j}}=-\frac{1}{2}\left(\frac{w(z)+\nu_{j}}{w(z)-\nu_{j}}+\frac{w(z)+\bar{\nu}_{j}}{w(z)-\bar{\nu}_{j}}\right)\frac{\partial\log
r}{\partial\lambda_{j}},\quad\nu_{j}:=e^{i\xi_{j}}.$ (6.1)

The driving functions $\xi_{j}=\xi_{j}(\\{\lambda_{i}\\})$ are real-valued, so
$|\nu_{j}|=1$, i.e., $\bar{\nu}_{j}=\nu_{j}^{-1}$. The compatibility
conditions of this system are

$\displaystyle{\frac{\partial}{\partial\lambda_{k}}\frac{\partial\log
w(z)}{\partial\lambda_{j}}=\frac{\partial}{\partial\lambda_{j}}\frac{\partial\log
w(z)}{\partial\lambda_{k}}}.$

In order to resolve these conditions, one should substitute equations (6.1)
and cancel all poles. A long but straightforward calculation shows that the
compatibility conditions are equivalent to the following system of the
Gibbons-Tsarev type:

$\left\\{\begin{array}[]{l}\displaystyle{\frac{\partial\nu_{j}}{\partial\lambda_{k}}=\frac{\nu_{j}}{2}\left(\frac{\nu_{k}+\nu_{j}}{\nu_{k}-\nu_{j}}+\frac{\bar{\nu}_{k}+\nu_{j}}{\bar{\nu}_{k}-\nu_{j}}\right)\frac{\partial\log
r}{\partial\lambda_{k}},}\\\ \\\ \displaystyle{\frac{\partial^{2}\log
r}{\partial\lambda_{j}\partial\lambda_{k}}=2\left(\frac{\nu_{j}\nu_{k}}{(\nu_{j}-\nu_{k})^{2}}+\frac{\nu_{j}\bar{\nu}_{k}}{(\nu_{j}-\bar{\nu}_{k})^{2}}\right)\frac{\partial\log
r}{\partial\lambda_{j}}\,\frac{\partial\log
r}{\partial\lambda_{k}}.}\end{array}\right.$ (6.2)

Now we are going to use the fact that each $\lambda_{j}$ is a function of the
times: $w(z;{\bf t})=w(z;\\{\lambda_{j}({\bf t})\\})$. Applying the chain rule
of differentiating, in the case of $N$-variable reduction we can rewrite
equation (3.7) of the dispersionless constrained Toda hierarchy in the form

$\sum_{j=1}^{N}D(\zeta)\lambda_{j}\cdot\partial_{\lambda_{j}}\log
w(z)=\sum_{j=1}^{N}\partial_{t_{0}}\lambda_{j}\cdot\partial_{\lambda_{j}}\log\left(\frac{w(\zeta)-w^{-1}(z)}{w(\zeta)-w(z)}\right).$
(6.3)

A rather long but straightforward calculation using the symmetric Löwner
equation (6.1) yields

$\begin{split}&\partial_{\lambda_{j}}\log\left(\frac{w(\zeta)-w^{-1}(z)}{w(\zeta)-w(z)}\right)\\\
&=-\frac{1}{2}\left(\frac{\nu_{j}}{w(\zeta)-\nu_{j}}+\frac{\bar{\nu}_{j}}{w(\zeta)-\bar{\nu}_{j}}\right)\left(\frac{w(z)+\nu_{j}}{w(z)-\nu_{j}}+\frac{w(z)+\bar{\nu}_{j}}{w(z)-\bar{\nu}_{j}}\right)\partial_{\lambda_{j}}\log
r.\\\
&=\left(\frac{\nu_{j}}{w(\zeta)-\nu_{j}}+\frac{\bar{\nu}_{j}}{w(\zeta)-\bar{\nu}_{j}}\right)\frac{\partial\log
w(z)}{\partial\lambda_{j}}.\end{split}$

Therefore, if we introduce the dependence of the $\lambda_{j}$’s on the times
by means of the relations

$D(z)\lambda_{j}=\left(\frac{\nu_{j}}{w(z)-\nu_{j}}+\frac{\bar{\nu}_{j}}{w(z)-\bar{\nu}_{j}}\right)\partial_{t_{0}}\lambda_{j},$
(6.4)

the equation (6.3) will be satisfied identically.

As it follows from (5.11), equation (6.4) is equivalent to an infinite system
of partial differential equations of hydrodynamic type:

$\frac{\partial\lambda_{j}}{\partial
t_{k}}=\Bigl{(}\phi_{j,k}(\\{\lambda_{i}\\})+\overline{\phi_{j,k}(\\{\lambda_{i}\\})}\,\Bigr{)}\frac{\partial\lambda_{j}}{\partial
t_{0}},\quad\phi_{j,k}=\nu_{j}B^{\prime}_{k}(\nu_{j}),$ (6.5)

where $B^{\prime}_{k}(w)\equiv\partial_{w}B_{k}(w)$. The generating function
of $\phi_{j,k}$’s is

$\begin{array}[]{c}\displaystyle{\sum_{k\geq
1}\Bigl{(}\phi_{j,k}(\\{\lambda_{i}\\})+\overline{\phi_{j,k}(\\{\lambda_{i}\\})}\,\Bigr{)}\frac{z^{-k}}{k}=\frac{\nu_{j}}{w(z)-\nu_{j}}+\frac{\bar{\nu}_{j}}{w(z)-\bar{\nu}_{j}}}\\\
\\\
\displaystyle{:=Q\Bigl{(}w(z,\\{\lambda_{i}\\}),\nu_{j}(\\{\lambda_{i}\\})\Bigr{)}.}\end{array}$
(6.6)

Now we are going to show that the system (6.5) is consistent and can be solved
by Tsarev’s generalized hodograph method [28]. It can be directly verified
that the compatibility condition of the system (6.5) is

$\frac{\partial_{\lambda_{j}}{\rm Re}\,\phi_{i,n}}{{\rm Re}\,\phi_{j,n}-{\rm
Re}\,\phi_{i,n}}=\frac{\partial_{\lambda_{j}}{\rm
Re}\,\phi_{i,n^{\prime}}}{{\rm Re}\,\phi_{j,n^{\prime}}-{\rm
Re}\,\phi_{i,n^{\prime}}}\quad\mbox{for all $i\neq j,\,n,\,n^{\prime}$}.$
(6.7)

In other words, the condition is that

$\Gamma_{ij}:=\frac{\partial_{\lambda_{j}}{\rm Re}\,\phi_{i,n}}{{\rm
Re}\,\phi_{j,n}-{\rm Re}\,\phi_{i,n}}$ (6.8)

does not depend on $n$. It is easy to see that this is equivalent to the
$z$-independence of the ratio

$\frac{\partial_{\lambda_{j}}Q(w(z),\nu_{i})}{Q(w(z),\nu_{j})-Q(w(z),\nu_{i})},$

where $Q$ is the generating function (6.6). If this holds, then

$\Gamma_{ij}=\frac{\partial_{\lambda_{j}}Q(w(z),\nu_{i})}{Q(w(z),\nu_{j})-Q(w(z),\nu_{i})}.$
(6.9)

A direct calculation which makes use of the symmetric Löwner equation (6.1)
and the Gibbons-Tsarev equations (6.2) gives

$\Gamma_{ij}=\left(\frac{\nu_{i}\nu_{j}}{(\nu_{i}-\nu_{j})^{2}}+\frac{\nu_{i}\nu_{j}}{(\nu_{i}\nu_{j}-1)^{2}}\right)\partial_{\lambda_{j}}\log
r.$ (6.10)

Let $R_{i}=R_{i}(\\{\lambda_{j}\\})$ ($i=1,\ldots,N$) satisfy the system of
equations

$\frac{\partial R_{i}}{\partial\lambda_{j}}=\Gamma_{ij}(R_{j}-R_{i}),\qquad
i,j=1,\ldots,N,\quad i\neq j,$ (6.11)

where $\Gamma_{ij}$ is defined in (6.10) (for $N=1$ this condition is void).
We claim that the system (6.11) is compatible in the sense of Tsarev [28]. To
see this, we note that $\Gamma_{ij}$ can be expressed as

$\Gamma_{ij}=\frac{1}{2}\,\partial_{\lambda_{j}}\log g_{i},\quad
g_{i}=\frac{\partial\log r}{\partial\lambda_{i}}.$ (6.12)

This directly follows from the Gibbons-Tsarev equations (6.2). It is then
obvious that

$\frac{\partial\Gamma_{ij}}{\partial\lambda_{k}}=\frac{\partial\Gamma_{ik}}{\partial\lambda_{j}},\qquad
i\neq j\neq k,$ (6.13)

which are Tsarev’s compatibility conditions. This means that the system (6.5)
is semi-Hamiltonian. The main geometric object associated with a semi-
Hamiltonian system is a diagonal metric. The quantities $g_{i}=g_{ii}$ are
components of this metric while $\Gamma_{ij}=\Gamma_{ij}^{i}$ are the
corresponding Christoffel symbols. Moreover, from (6.12) it is clear that the
metric $g_{i}$ is of Egorov type, i.e.,

$\frac{\partial g_{i}}{\partial\lambda_{k}}=\frac{\partial
g_{k}}{\partial\lambda_{i}}.$ (6.14)

Assume that $R_{i}$ satisfy the system (6.11). Then the same argument as in
the proof of Theorem 10 of Tsarev’s paper [28] shows that if $\lambda_{i}$ is
defined implicitly by the hodograph relations

$t_{0}+2{\rm Re}\sum_{k\geq
1}t_{k}\phi_{i,k}(\\{\lambda_{j}\\})=R_{i}(\\{\lambda_{j}\\}),$ (6.15)

then $\lambda_{j}$ satisfy (6.5).

We have found sufficient conditions for $N$-variable diagonal reductions of
the dispersionless constrained Toda hierarchy. The reduction is given by a
system of $N$ symmetric Löwner equations (6.1) for a function
$w(z,\lambda_{1},\ldots,\lambda_{N})$ supplemented by a diagonal system of
hydrodynamic type (6.5) for the variables $\lambda_{j}$, $j=1,\ldots,N$.

##  7 The symmetric radial Löwner equation

In this section we clarify the geometric meaning of the symmetric radial
Löwner equation (5.7). We will sketch the derivation of a differential
equation obeyed by the conformal map of the unit disk with two symmetric slits
to the exterior of the unit which turns out to be the symmetric radial Löwner
equation. Rigorous proof is in Appendix A.

By $\mathbb{U}$ we denote the unit disk $|z|<1$ and
$\mathbb{U}^{c}=\mathbb{C}\smallsetminus\mathbb{U}$ its compliment. Let
$\Gamma:[0,\infty)\to\mathbb{U}^{c}$ be a simple curve with no self-
intersections in the upper half plane with the condition $|\Gamma(0)|=1$ and
$\bar{\Gamma}$ be the mirror symmetric curve with respect to the real axis:
$\bar{\Gamma}(t)=\overline{\Gamma(t)}$. The curves $\Gamma$ and $\bar{\Gamma}$
do not intersect, as $\Gamma(t)$ is in the upper half plane. Let $\Gamma_{t}$
be the arc of the curve $\Gamma$ corresponding to the values of the parameter
from $0$ to $t$: $\Gamma_{t}:[0,t]\to\mathbb{U}^{c}$, and $\bar{\Gamma}_{t}$
be the symmetric arc in the lower half plane.

Let $g(z,t)$ be the univalent conformal map from the domain $\mathbb{D}_{t}=$
(interior of $\mathbb{U}^{c}$)
$\smallsetminus(\Gamma_{t}\cup\bar{\Gamma}_{t})$ to
$\overset{\circ}{\mathbb{U}^{c}}=$ (interior of $\mathbb{U}^{c}$) normalized
by the condition

$g(z,t)=z/r(t)+O(1),\quad z\to\infty,\quad r(t)\in\mathbb{R}_{+}.$ (7.1)

The Riemann mapping theorem ensures that such a map exists and is unique. The
quantity $r(t)$ is called the conformal radius of the domain $\mathbb{D}_{t}$
relative to infinity.

Figure 1: Conformal map $g(z,t)$.

Our goal is to derive a differential equation for the function $g(z,t)$ in the
variable $t$. According to our convention, $g(z,0)=z$, which is going to be
the initial condition for the differential equation. Note that since the
domain $\mathbb{D}_{t}$ is symmetric and $g(z,t)$ is normalised as (7.1)
($r(t)\in\mathbb{R}_{+}$), the function $g(z,t)$ enjoys the property

$g(z,t)=\overline{g(\bar{z},t)}.$ (7.2)

(See Lemma A.1.)

The main technical tool is the complex Poisson formula (or the Schwarz
integral formula) which is a version of the Cauchy integral formula. Given a
bounded continuous function $f(z)$ in $\mathbb{U}^{c}$, holomorphic in its
interior $\overset{\circ}{\mathbb{U}^{c}}$ (thus holomorphic also at
$z=\infty$ by Riemann’s removable singularity theorem), the complex Poisson
formula reads

$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\rm
Re}\,f(e^{i\theta})\frac{z+e^{i\theta}}{z-e^{i\theta}}\,d\theta+i\,{\rm
Im}\,f(\infty).$ (7.3)

We define a map $h(z;s,t)$ ($0<s<t$) for $z\in\mathbb{U}^{c}$ by

$h(z;s,t)=g(g^{-1}(z,t),s)=\frac{r(t)}{r(s)}\,z+O(1),$ (7.4)

where $g^{-1}$ is the function inverse to $g$. (See Figure 2.)

Figure 2: Conformal map $h(z;s,t)$.

We want to apply the complex Poisson formula to the function
$\log(h(z;s,t)/z)$. Hence, hereafter we assume that $g^{-1}(z,t)$ and hence
$h(z;s,t)$ are continuously extended to $\mathbb{U}^{c}$, the closure of the
original domain of definition $\overset{\circ}{\mathbb{U}^{c}}$. (In Appendix
A we shall see that such extension is really possible for certain class of
functions.)

We get:

$\log\frac{h(z;s,t)}{z}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log|h(e^{i\theta});s,t)|\,\frac{z+e^{i\theta}}{z-e^{i\theta}}\,d\theta.$
(7.5)

In particular, tending $z\to\infty$, we obtain the following corollary of
(7.5):

$\log r(t)-\log
r(s)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log|h(e^{i\theta});s,t)|d\theta.$ (7.6)

Substituting $z\mapsto g(z,t)$ in (7.5), we can write:

$\log\frac{g(z,s)}{g(z,t)}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log|h(e^{i\theta});s,t)|\,\frac{g(z,t)+e^{i\theta}}{g(z,t)-e^{i\theta}}\,d\theta.$
(7.7)

The property (7.2) implies the similar property of the function $h(z;s,t)$:
$h(z;s,t)=\overline{h(\bar{z};s,t)}$ which allows one to rewrite the relations
(7.6), (7.7) as follows:

$\log r(t)-\log
r(s)=\frac{1}{\pi}\int_{0}^{\pi}\log|h(e^{i\theta});s,t)|d\theta,$ (7.8)
$\log\frac{g(z,s)}{g(z,t)}=\frac{1}{2\pi}\int_{0}^{\pi}\log|h(e^{i\theta});s,t)|\left(\frac{g(z,t)+e^{i\theta}}{g(z,t)-e^{i\theta}}+\frac{g(z,t)+e^{-i\theta}}{g(z,t)-e^{-i\theta}}\right)d\theta.$
(7.9)

Let the image of the tip of the curve $\Gamma_{t}$ be $e^{i\xi(t)}$:
$g(\Gamma(t),t)=e^{i\xi(t)}$. Then the image of the tip of the curve
$\bar{\Gamma}_{t}$ is $e^{-i\xi(t)}$. Note that $\log|h(e^{i\theta};s,t)|=0$
if $e^{i\theta}$ is mapped by $h$ to the boundary of $\mathbb{U}^{c}$, i.e. to
a point on the unit circle. Accordingly, $\log|h(e^{i\theta};s,t)|\neq 0$ if
the point $e^{i\theta}$ lies in a neighborhood of $e^{i\xi(t)}$ (in an arc of
the unit circle containing the point $e^{i\xi(t)}$) or in a neighborhood of
$e^{-i\xi(t)}$. If $s\to t$ from below, this neighborhood shrinks to a point
($e^{i\xi(t)}$ or $e^{-i\xi(t)}$). Therefore, equations (7.8), (7.9) imply the
following differential equation by the mean value theorem444Of course in this
argument derivative $\partial_{t}$ in (7.10) should be interpreted as the left
derivative, but we can show the same equation for the right derivative, too.
See Appendix A.:

$\partial_{t}\log
g(z,t)=-\frac{1}{2}\left(\frac{g(z,t)+e^{i\xi(t)}}{g(z,t)-e^{i\xi(t)}}+\frac{g(z,t)+e^{-i\xi(t)}}{g(z,t)-e^{-i\xi(t)}}\right)\partial_{t}\log
r,$ (7.10)

which is the symmetric radial Löwner equation (5.7) ($t\to\lambda$, $g(z,t)\to
w(z)$). We see that solutions to this equation such that $g(z,0)=z$ are
functions which conformally map the exterior of the unit circle with two non-
intersecting symmetric slits $\Gamma_{t}$ (in the upper half plane) and
$\bar{\Gamma}_{t}$ (in the lower half plane) to the exterior of the unit
circle.

##  8 Conclusion

In this paper we have introduced the dispersionless version of the constrained
Toda hierarchy. It was formulated in two equivalent ways: a) by means of the
Lax formalism and b) by means of the Hirota-like equation for the $F$-function
(the dispersionless limit of logarithm of the tau-function). The geometric
meaning of the dispersionless constrained Toda hierarchy was clarified. We
have shown that solutions to the hierarchy describe conformal maps of domains
symmetric under reflection with respect to the real axis to the reference
domain (the unit disk).

We have also studied finite-dimensional reductions of the dispersionless
constrained Toda hierarchy. It was shown that the consistency condition for
one-variable reduction is a differential equation of the Löwner type which we
call the symmetric radial Löwner equation. The geometric meaning of the latter
was clarified. We have shown that solutions to this equation are conformal
maps of the exterior of the unit circle with two symmetric curved slits to the
exterior of the unit circle.

##  Appendix A Appendix: Proof of the symmetric radial Löwner equation

In this appendix we derive the symmetric radial Löwner equation for an
evolution family, a two-parameter family of conformal mappings, rigorously. In
particular extendability of the maps is discussed here. The last part of the
proof is almost the same as the arguments in Section 7 but with some omitted
details.

Assume that a Jordan curve $\gamma$ is in the upper half plane:

$\gamma:[0,+\infty)\to\mathbb{C},\qquad\mathrm{Im}\,\gamma(t)>0,$ (A.1)

and denote its complex conjugate by $\bar{\gamma}$:

$\bar{\gamma}(t):=\overline{\gamma(t)}.$ (A.2)

Let $D_{t}$ ($t\geq 0$) be the following domain, which is symmetric with
respect to the real axis:

$D_{t}:=\mathbb{C}\smallsetminus\bigl{(}\gamma([t,+\infty))\cup\bar{\gamma}([t,+\infty))\bigr{)}.$
(A.3)

As in Section 7, the following lemma is essential.

######  Lemma A.1.

Let $D$ be a simply-connected domain in $\mathbb{C}$ symmetric with respect to
the real axis: $z\in D\Longleftrightarrow\bar{z}\in D$. For simplicity we
assume that $0\in D$. Then the holomorphic bijection $f:\mathbb{U}\to D$
normalised by $f(0)=0$ and $f^{\prime}(0)>0$, the existence of which is
guaranteed by Riemann’s mapping theorem, satisfies

$f(\bar{z})=\overline{f(z)}$ (A.4)

for $z\in D$.

###### Proof.

Note that $\tilde{f}(z):=\overline{f(\bar{z})}$ is also a holomorphic
bijection between $\mathbb{U}$ and $D$, satisfying the normalisation
conditions $\tilde{f}(0)=0$, $\tilde{f}^{\prime}(0)>0$. Since the Riemann
mapping theorem claims that such a mapping is unique, $\tilde{f}=f$, which
proves (A.4). ∎

We denote the normalised conformal mapping from the unit open disk
$\mathbb{U}$ to $D_{t}$ by $f_{t}$:

$f_{t}:\mathbb{U}\to D_{t},\qquad f_{t}(0)=0,\ f^{\prime}_{t}(0)>0.$ (A.5)

Because of Lemma A.1 this mapping function satisfies

$f_{t}(\bar{z})=\overline{f_{t}(z)}.$ (A.6)

The argument in §3.3 [11] or §3.2, Chapter 3 of [29] can be applied to the
family of domains $\\{D_{t}\\}$ and we can reparametrise it so that $f_{t}$
satisfies $f_{t}^{\prime}(0)=e^{t}$:

$f_{t}(z)=e^{t}z+a_{2}(t)\,z^{2}+a_{3}(t)\,z^{3}+\dotsb.$ (A.7)

(The choice $f_{t}^{\prime}(0)=e^{t}$ is just for simplicity. We can take any
smooth positive increasing function $r(t)$ as in Section 7 instead of
$e^{t}$.) We call the family $\\{f_{t}(z)\\}_{t\geq 0}$ the symmetric Löwner
chain.

For non-negative real numbers $s$ and $t$ satisfying $0\leq s<t<+\infty$ we
define the symmetric evolution family $\omega_{s,t}:\mathbb{U}\to\mathbb{U}$
by $\omega_{s,t}:=f_{t}^{-1}\circ f_{s}$, which satisfies $\omega_{s,t}(0)=0$,
$f_{s}=f_{t}\circ\omega_{s,t}$,

$\omega_{s,t}(z)=e^{s-t}z+b_{2}(t)\,z^{2}+b_{3}(z)\,z^{3}+\dotsb,$ (A.8)

and is univalent. Each $\omega_{s,t}$ also satisifes the conditions of Lemma
A.1:

$\omega_{s,t}(\bar{z})=\overline{\omega_{s,t}(z)}.$ (A.9)

######  Theorem A.1.

For a fixed $s\geq 0$ and $z\in\mathbb{U}$ the function
$\omega(t)=\omega_{s,t}(z)$ satisfies an ordinary differential equation

$\frac{d\omega}{dt}=-\frac{\omega}{2}\left(\frac{e^{i\theta(t)}+\omega}{e^{i\theta(t)}-\omega}+\frac{e^{-i\theta(t)}+\omega}{e^{-i\theta(t)}-\omega}\right)$
(A.10)

on $t\in[s,\infty)$ with the initial value condition $w(s)=z$. Here
$\theta(t)$ is a real continuous function on $[0,+\infty)$.

By changing the variables $t$, $z$ and $\omega_{s,t}$ by $\lambda:=-t$,
$\zeta:=z^{-1}$, $w(\zeta,\lambda):=1/\omega_{s,t}(1/\zeta)$, we obtain a
conformal mapping from $\mathbb{C}\smallsetminus\mathbb{U}$ to
$\mathbb{C}\smallsetminus\mathbb{U}$, which satisfies

$\frac{dw}{d\lambda}=-\frac{w}{2}\left(\frac{w+e^{-i\theta(\lambda)}}{w-e^{-i\theta(\lambda)}}+\frac{w+e^{i\theta(\lambda)}}{w-e^{i\theta(\lambda)}}\right),$
(A.11)

which is nothing but (5.7).

###  A.1 Behaviour of functions on the boundaries

In this subsectin we prove extendability of $\omega_{s,t}$ to the boundary of
$\mathbb{U}$. The variable $t$ is fixed. The argument almost follows that in
§3.2, Chapter 3 of [29], which is an improved version of the argument in [11].

If $0\leq s<t$, $D_{t}\smallsetminus
D_{s}=\gamma([s,t))\cup\bar{\gamma}([s,t))$. Hence by (A.9) the image of
$\omega_{s,t}$ is

$\omega_{s,t}(\mathbb{U})=\mathbb{U}\smallsetminus(\alpha_{s,t}\cup\overline{\alpha_{s,t}}),$

where $\alpha_{s,t}:=f_{t}^{-1}\bigl{(}\gamma([s,t))\bigr{)}$ and
$\overline{\alpha_{s,t}}$ is its complex conjugate.

Let us show that each arc $\alpha_{s,t}$ has an endpoint $\lambda(t)$ on
$\partial\mathbb{U}$, $\displaystyle\lambda(t)=\lim_{s\nearrow
t}f_{t}^{-1}(\gamma(s))$, which does not depend on $s\in[0,t)$.

We apply the Carathéodory extension theorem by opening the slit

$\gamma([t,\infty))\cup\bar{\gamma}([t,\infty])\cup\\{\infty\\}$

in the Riemann sphere $\widehat{\mathbb{C}}$ by square root555In the
derivation of Löwner’s equation in §3.3 of [11] Carathéodory’s theorem is
applied to a domain whose boundary is not a Jordan curve. In [29] this gap is
filled by opening the slit by square root. We use this idea here.. We take a
branch of

$\zeta=F(w):=\sqrt{\frac{1-w/\gamma(t)}{1-w/\bar{\gamma}(t)}}$ (A.12)

satisfying $F(0)=1$. Its inverse map is

$w=G(\zeta):=\gamma(t)\,\bar{\gamma}(t)\frac{1-\zeta^{2}}{\bar{\gamma}(t)-\gamma(t)\,\zeta^{2}}.$
(A.13)

Endpoints of the slit are mapped by $F$ to $F(\gamma(t))=0$ and
$F(\bar{\gamma}(t))=\infty$. The image of the slit is a Jordan closed curve
$\Gamma^{+}\cup\\{0\\}\cup\Gamma^{-}\cup\\{\infty\\}$ in
$\widehat{\mathbb{C}}$, where $\Gamma^{\pm}$ are two connected components of
$F\bigl{(}\gamma((t,+\infty))\bigr{)}$ (Figure 3).

Figure 3: Extension of $F\circ f_{t}$.

Applying the Carathéodory extension theorem to $F\circ f_{t}$, we obtain a
homeomorphism $h:\overline{\mathbb{U}}\to\overline{F(D_{t})}$ which is
holomorphic in $\mathbb{U}$. Thus we obtain an extension of $f_{t}$ on
$\overline{\mathbb{U}}$: $f_{t}=G\circ
h:\overline{\mathbb{U}}\to\overline{D_{t}}=\widehat{\mathbb{C}}$. (We use the
same notation for the extended map as the original one.)

Since $f_{t}=F^{-1}\circ h$ on $\mathbb{U}$, we have

$f_{t}^{-1}(\gamma(s))=h^{-1}(F(\gamma(s))),\quad
f_{t}^{-1}(\bar{\gamma}(s))=h^{-1}(F(\bar{\gamma}(s)))$

for $0\leq s<t$. Hence, when $s$ approaches to $t$ from below, there exists
limits,

$\displaystyle\lambda(t)$ $\displaystyle:=\lim_{s\nearrow
t}f_{t}^{-1}(\gamma(s))=\lim_{s\nearrow
t}h^{-1}(F(\gamma(s)))=h^{-1}(F(\gamma(t)))=h^{-1}(0),$ (A.14)
$\displaystyle\bar{\lambda}(t)$ $\displaystyle:=\lim_{s\nearrow
t}f_{t}^{-1}(\bar{\gamma}(s))=\lim_{s\nearrow
t}h^{-1}(F(\bar{\gamma}(s)))=h^{-1}(F(\bar{\gamma}(t)))=h^{-1}(\infty),$
(A.15)

and, due to (A.2) and (A.6), they are complex conjugate to each other,

$\bar{\lambda}(t)=\overline{\lambda(t)}.$ (A.16)

The proof of (right/left) continuity of $\lambda(t)$ is the same as that in
§3.3 of [11].

As in Figure 4 we denote the counterclockwise arc from $\lambda(t)$ to
$\bar{\lambda}(t)$ by $I^{+}=I^{+}_{t}$ and the other arc by
$I^{-}=I^{-}_{t}$: $h(I^{\pm})=\Gamma^{\pm}$. Restrictions of
$f_{t}:\bar{\mathbb{U}}\to\widehat{\mathbb{C}}$ on $I^{\pm}$,
$f_{t}^{\pm}:I^{\pm}\to\gamma([t,+\infty))\cup\\{\infty\\}\cup\bar{\gamma}([t,+\infty))$,
are homeomorphisms. Therefore for any $u\in(t,+\infty)$ the subset
$J_{t,u}:=J_{t,u}^{+}\cup J_{t,u}^{-}$ of $\partial\mathbb{U}$, where
$J^{\pm}_{t,u}:=(f_{t}^{\pm})^{-1}\bigl{(}\gamma([t,u))\bigr{)}$, is an open
arc of $\partial\mathbb{U}$.

Figure 4: Arcs $J_{t,u}$ and $\bar{J}_{t,u}$.

Similarly, the subset $\bar{J}_{t,u}:=\bar{J}_{t,u}^{+}\cup\bar{J}_{t,u}^{-}$
of $\partial\mathbb{U}$, where
$\bar{J}^{\pm}_{t,u}:=(f_{t}^{\pm})^{-1}\bigl{(}\bar{\gamma}([t,u))\bigr{)}$,
is an open arc of $\partial\mathbb{U}$ and, because of (A.6),
$\bar{J}_{t,u}^{\pm}=\overline{J_{t,u}^{\pm}}$,
$\bar{J}_{t,u}=\overline{J_{t,u}}$.

Then $f_{t}(J_{t,u})=\gamma([t,u))$ and
$f_{t}(\bar{J}_{t,u})=\bar{\gamma}([t,u))$ and, if $0\leq t<u$,

$I_{t}^{\pm}\smallsetminus(J_{t,u}^{\pm}\cup\bar{J}_{t,u}^{\pm})=f_{t}^{-1}\bigl{(}\gamma([u,\infty))\cup\bar{\gamma}([u,\infty))\bigr{)}\cap
I_{t}^{\pm}.$

Therefore, if we define $\omega_{t,u}$ on the boundary $\partial\mathbb{U}$ as

$\omega_{t,u}(z):=\begin{cases}f_{u}^{-1}\bigl{(}f_{t}(z)\bigr{)},&(z\in\mathbb{U}\cup
J_{t,u}\cup\bar{J}_{t,u}),\\\
(f_{u}^{\pm})^{-1}\bigl{(}f_{t}(z)\bigr{)},&(z\in
I_{t}^{\pm}\smallsetminus(J_{t,u}\cup\bar{J}_{t,u})),\end{cases}$ (A.17)

it is a continuous map $\omega_{t,u}:\bar{\mathbb{U}}\to\bar{\mathbb{U}}$.
This is the desired extension of the symmetric evolution family
$\omega_{t,u}(z)$.

The images of subsets $\mathbb{U}$, $J_{t,u}$ and $\bar{J}_{t,u}$ of
$\bar{\mathbb{U}}$ are

$\omega_{t,u}(\mathbb{U})=\mathbb{U}\smallsetminus(\alpha_{t,u}\cup\bar{\alpha}_{t,u}),\quad\omega_{t,u}(J_{t,u})=\alpha_{t,u},\quad\omega_{t,u}(\bar{J}_{t,u})=\bar{\alpha}_{t,u},$

respectively, as shown in Figure 5.

Figure 5: Map $\omega_{t,u}$ and arcs $\alpha_{t,u}$, $\bar{\alpha}_{t,u}$.

###  A.2 Symmetric radial Löwner equation for $\\{\omega_{s,t}(z)\\}$.

We add some details in derivation of the symmetric radial Löwner equation
discussed in Section 7. (Recall that in this appendix $r(t)$ is normalized as
$r(t)=e^{t}$ for simplicity.) We follow the argument in §3.2, Chapter 3 of
[29].

As $\omega_{t,u}(z)=e^{t-u}z+O(z^{2})$, we can take a branch of

$h(z)=h(z;t,u):=\log\frac{\omega_{t,u}(z)}{z},$ (A.18)

such that $h(0)=t-u$. This function is holomorphic in $\mathbb{U}$ and
continuous on its closure $\bar{\mathbb{U}}$. Since
$\omega_{t,u}(\partial\mathbb{U}\smallsetminus(J_{t,u}\cup\bar{J}_{t,u}))\subset\partial\mathbb{U}$,
$\mathrm{Re}\,h(z)=0$, if
$z\in\partial\mathbb{U}\smallsetminus(J_{t,u}\cup\bar{J}_{t,u})$, and since
$\omega_{t,u}(J_{t,u}\cup\bar{J}_{t,u})\subset\mathbb{U}$,
$\mathrm{Re}\,h(z)<0$, if $z\in J_{t,u}\cup\bar{J}_{t,u}$. Hence, applying the
Schwarz integral formula for the unit disk666Recall that formula (7.3) is for
functions outside of the unit disk.,

$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{Re}\,f(e^{i\theta})\frac{e^{i\theta}+z}{e^{i\theta}-z}d\theta+i\mathrm{Im}\,f(0),\qquad(z\in\mathbb{U}),$

to $h(z)$, we have

$h(z)=\frac{1}{2\pi}\left(\int_{a}^{b}+\int_{-b}^{-a}\mathrm{Re}\,h(e^{i\theta})\frac{e^{i\theta}+z}{e^{i\theta}-z}d\theta\right),$
(A.19)

where $a$ and $b$ are real numbers such that $e^{ia}$ and $e^{ib}$ are
endpoints of $J_{t,u}$ and, correspondingly, $e^{-ib}$ and $e^{-ia}$ are
endpoints of $\bar{J}_{t,u}$. Note that $h(0)=t-u\in\mathbb{R}$ and therefore
$\mathrm{Im}\,h(0)=0$.

As in Section 7, because of the symmetry $h(\bar{z})=\overline{h(z)}$ the
above formula is rewritten as

$h(z)=\frac{1}{\pi}\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})K(e^{i\theta},z)d\theta,$
(A.20)

where $K(\zeta,z)$ is the function defined by $\displaystyle
K(z,\zeta):=\frac{1}{2}\left(\frac{\zeta+z}{\zeta-z}+\frac{\bar{\zeta}+z}{\bar{\zeta}-z}\right)$.
Substituting $\omega_{s,t}(z)$ for $z$ in (A.20) and using
$\omega_{t,u}(\omega_{s,t}(z))=\omega_{s,u}(z)$, we have

$\log\frac{\omega_{s,u}(z)}{\omega_{s,t}(z)}=\frac{1}{\pi}\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})K_{R}(e^{i\theta},\omega_{s,t}(z))d\theta+\frac{i}{\pi}\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})K_{I}(e^{i\theta},\omega_{s,t}(z))d\theta,$
(A.21)

where $K_{R}(\zeta,z):=\mathrm{Re}\,K(\zeta,z)$ and
$K_{I}(\zeta,z):=\mathrm{Im}\,K(\zeta,z)$. From the mean value theorem of
integrals it follows that there exist $\xi$ and $\eta$ in the interval $(a,b)$
such that

$\displaystyle\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})K_{R}(e^{i\theta},\omega_{s,t}(z))d\theta$
$\displaystyle=K_{R}(e^{i\xi},\omega_{s,t}(z))\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})d\theta,$
$\displaystyle\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})K_{I}(e^{i\theta},\omega_{s,t}(z))d\theta$
$\displaystyle=K_{I}(e^{i\eta},\omega_{s,t}(z))\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})d\theta.$

Hence by dividing (A.21) by $\displaystyle
t-u=h(0)=\frac{1}{\pi}\int_{a}^{b}\mathrm{Re}\,h(e^{i\theta})\,d\theta,$ we
obtain

$\frac{\log\omega_{s,u}(z)-\log\omega_{s,t}(z)}{u-t}=-\bigl{(}K_{R}(e^{i\xi},\omega_{s,t}(z))+i\,K_{I}(e^{i\eta},\omega_{s,t}(z))\bigr{)}.$

When $u\searrow t$, $J_{t,u}$ contracts to $\lambda(t)$ (cf. Figure 4) and
therefore both of $e^{i\xi}$ and $e^{i\eta}$ tend to $\lambda(t)$. Thus

$\frac{\partial}{\partial
t}\log\omega_{s,t}(z)=-K(\lambda(t),\omega_{s,t}(z)),$ (A.22)

where the left hand side is regarded as the right derivative. Similarly the
limit $t\nearrow u$ gives (A.22) as the left derivative, because $J_{t,u}$
contracts to $\lambda(u)$ in this limit. Thus we have proved the symmetric
radial Löwner equation (A.10) for $\omega(t)=\omega_{s,t}(z)$,
$e^{i\theta(t)}=\lambda(t)$.

## Acknowledgements

We are grateful to Ikkei Hotta for his detailed comments on complex analysis.
The results of T.T. (Sections 6,7 and Appendix) is an output of a research
project implemented as part of the Basic Research Program at the National
Research University Higher School of Economics. The work of A.Z. (Sections
2–5) was performed at the Steklov Mathematical Institute of Russian Academy of
Sciences, Moscow, and was supported by the Russian Science Foundation under
grant 19-11-00062.

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# Deep Learning-Based Weather-Related Power Outage Prediction with Socio-
Economic and Power Infrastructure Data

Xuesong Wang, Nina Fatehi, Caisheng Wang*, Masoud H. Nazari
Department of Electrical and Computer Engineering, Wayne State University,
Detroit, MI, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>*Corresponding author.

###### Abstract

This paper presents a deep learning-based approach for hourly power outage
probability prediction within census tracts encompassing a utility company’s
service territory. Two distinct deep learning models, conditional Multi-Layer
Perceptron (MLP) and unconditional MLP, were developed to forecast power
outage probabilities, leveraging a rich array of input features gathered from
publicly available sources including weather data, weather station locations,
power infrastructure maps, socio-economic and demographic statistics, and
power outage records. Given a one-hour-ahead weather forecast, the models
predict the power outage probability for each census tract, taking into
account both the weather prediction and the location’s characteristics. The
deep learning models employed different loss functions to optimize prediction
performance. Our experimental results underscore the significance of socio-
economic factors in enhancing the accuracy of power outage predictions at the
census tract level.

###### Index Terms:

deep learning, power infrastructure, power outage, prediction, socio-economic

##  I Introduction

Modern societies heavily rely on the power grid to supply continuous power to
consumers. Power outages can cause serious and enormous costs. Extreme weather
and natural disasters, such as hurricanes, wildfires, and floodings, are
inevitable, which disturb the normal operation of power systems. According to
[1], 44%-78% of the reported power grid outages are weather-related, and the
cost estimates from storm-related outages to the U.S. economy are between $20
and $55 billion annually. Therefore, rapid response to such events is crucial.
The response includes preparing for and adapting to changes as well as quick
recovery and restoration from major disruptions [2]. Therefore, there is a
significant need to accurately forecast the extreme event impacts on the power
grid so that utility crews can be dispatched during or even before the events
and that long-term hardening approaches can be carried out in less resilient
communities.

Deep learning (DL) based models have also been proposed to predict power
outages. A Recurrent Neural Network (RNN) is developed in [3] to predict
outage duration by capturing temporal patterns in the outage data during
extreme weather events. Long Short-Term Memory (LSTM) models are also
developed to predict power load and power outage probability [4]. In [5],
authors compare the performances of different classification models for power
outage prediction, e.g., Random Forest, Multi-Layer Perceptron (MLP), Support
Vector Machine, K-Nearest Neighbor (KNN), AdaBoost, and Quadratic Discriminant
Analysis models. The authors of [6] combine Convolutional Neural Network (CNN)
and LSTM to predict outages for 10 randomly selected counties across New York
State.

Recently, conditional models have attracted much attention in AI Generated
Content (AIGC) [7, 8]. In conditional models, a condition is given to the
model to modulate the base features. In the weather-related power outage
prediction, weather data is the base input, and the characteristics of the
location are the condition [9, 10]. Different local conditions, e.g.,
infrastructure age and soil conditions, should lead to various outage
probabilities under the same weather condition.

The inputs to the models in the power outage prediction context usually
include weather information, power system details, environmental conditions,
and demographic data [9]. In [11], wind speed and the distance of each
component from the center of hurricane are used. A derived resiliency index
for each component is utilized in [12]. To predict power load, historical
power load data are utilized in [4]. To study the socio-economic impacts on
electricity-vulnerable population groups, e.g., household income and racial
demographics, authors of [13] designed an interactive tool to collect, analyze
and predict power outages in New York City. In power outage prediction,
various targets are employed, such as power outage occurrence (outage or non-
outage) [5], number of customers who lose power [9] and the ratio of power
outage customers over the population [10]. Sparsity or unbalanced data is an
issue in power outage prediction. To address it, authors of [14] propose a
Majority Under-Sampling and Minority Over-Sampling strategy to balance the
non-outage and outage samples.

In this paper, we consider two deep learning models, an unconditional MLP and
a conditional MLP, to predict the number of customers who lose power in an
hourly resolution, normalized by the population of the census tracts. We show
the effectiveness of the conditional model in power outage predictions. For
the input data, we consider not only weather features, but also the
characteristics of the communities (census tracts) in the DTE Energy service
territory and show how socio-economic factors impact the power outage
predictions. We also propose using cross-entropy loss function and assigning a
large weight to the outage class to mitigate the class imbalance issue in the
dataset and compare the performance of the weighted cross entropy loss and an
exponential loss on both models. Therefore, our main contribution can be
summarized as follows:

  * •

We have developed two deep learning models, an unconditional MLP and a
conditional MLP, to predict power outage probability with census tract
characteristics as the conditional input.

  * •

We have compared the performance of the conditional MLP with unconditional MLP
under two different loss functions: a weighted cross entropy loss and an
exponential loss.

  * •

We have investigated the impact of socio-economic factors as well as power
infrastructure features on power outage frequencies. The results show that
these factors effectively reduce the error rate in the proposed models.

The remaining sections are arranged as follows: In Section II, the data
collection and preprocessing are discussed. This is followed by the
introduction of the proposed power outage prediction method in section III.
Section IV presents the performance comparison between the conditional and
unconditional models and various input factors. In Section V, the conclusion
is summarized.

##  II Data Collection and Preprocessing

###  II-A Outage Data

The area under investigation encompasses the DTE’s service territory in
Michigan. Through DTE’s Outage Management System interface, outage data from
Feb. 27, 2023 to Sep. 30, 2023 were collected every 15 minutes. For each power
outage event, we take its first occurrence as the start time and the last
occurrence as the end time. As the spatial coverage and number of customers
may change during the event period, we take the largest spatial coverage and
the maximum number of customers during its period as its coverage and affected
customers, respectively. To align the power outage events to the census tract
level, a pairwise intersection operation is executed in ArcGIS Pro,
distributing the affected customers proportionally. The average power outage
duration in hours per capita for each census tract is shown in Fig. 1. After
removing all the power outage events affecting less than or equal to 1
customer, we count the number of customers that lost power in each hour in the
studied period. In total, 1,102 census tracts are selected, and for each
census tract, 5,184 hourly power outage probabilities are calculated. Note
that outages usually occur after storms or high wind gusts, resulting in the
sparsity of outages in the dataset, i.e., on average 26 out of 5184 hours are
outage hours for all census tracts.

In cases where the number of affected customers surpasses the population count
of the respective census tract, we assign a probability value of 1. Note that
while certain power outage events may not be solely attributed to weather
conditions, we retain them in the dataset for two primary reasons. Firstly,
the majority of identified event causes are weather-related, such as wind or
ice impacts. Secondly, a substantial number of events lack a definitive cause
within the records, as illustrated in Fig. 2. We consider the variability
present in the power outage data as a form of data augmentation and a
regularization mechanism for our model.

Figure 1: The average power outage duration in hours per capita for each
census tract.

Figure 2: Histogram of the causes of the power outage events.

###  II-B Weather and Weather Station Data

The historical hourly weather data are collected from Automated Surface
Observing System (ASOS).111Downloaded from
https://mesonet.agron.iastate.edu/ASOS/ We select 10 features from hourly
observations, including air temperature, relative humidity, pressure
altimeter, wind speed and direction, etc. Regarding temperature, humidity, and
air pressure data, missing values are addressed through temporal
interpolation. As for the remaining features, any missing values are filled
with zeros. In cases where observations are missing, we employ the average of
the available data from other weather stations. Subsequently, the data is
normalized by mean and standard deviation.

###  II-C Power Infrastructure Data

The power infrastructure data are collected from Open Street Map dataset [15],
as shown in Fig. 3. All nodes in the studied area that have attribute ”power”
are collected. The types of nodes include compensator, generator, insulator,
line, pole, portal, substation, switch, terminal, tower, and transformer. The
number of each feature is counted for census tracts using ArcGIS Pro. Note
that this dataset may not ensure data completeness. Nonetheless, we consider
the inherent noise within the data as a form of data augmentation and a
regularization technique for our model. Since the detailed topology of the
power distribution network is usually not publicly available, the Open Street
Map data [15] can serve as a good approximation of the true topology for
research purposes.

Figure 3: Power infrastructure components in the studied area.

###  II-D Socio-economic and Demographic Data

Even when exposed to identical weather conditions, census tracts with varying
levels of resilience are likely to encounter distinct degrees of power
outages. For each census tract, we collect the following data from the U.S.
Census Bureau:

  * •

Total population in 2020

  * •

Household income distribution estimated in 2021

  * •

Distribution of year structure built estimated in 2021

The household income and year structure built data include both estimations
and margins of error. Prior to normalizing the distribution with Softmax, we
employ a data augmentation technique during model training. This augmentation
involves generating a uniform distribution based on the mean and margin of
error for each data bin.

##  III Proposed Power Outage Prediction Method

We design two model structures, i.e., unconditional and conditional models.
For each model, we utilize two loss functions, i.e., exponential loss and
cross entropy loss. In the following subsections, we elaborate the proposed
models.

###  III-A Conditional MLP Structure

The conditional model comprises two primary branches, as illustrated in Fig.
4. The first branch takes weather features and the distances between weather
stations and the census tract as the input. These inputs are then processed
through several linear and non-linear layers. The second branch receives four
types of inputs: 1) household income distribution, 2) year structure built
distribution, 3) power infrastructure distribution, and 4) totals (including
the total population, total households, total houses, and total power
infrastructure components). These inputs are then processed through a series
of linear and non-linear layers. The output vector generated by the second
branch undergoes a transformation involving two sets of layers, resulting in a
scale vector and a bias vector. These two vectors are then applied to the
output of the first branch using (1), and the resulting output is further
processed by the final output layer. This design is driven by the concept
that, when confronted with identical weather conditions, the model should
adapt the influence of weather on the power outage probability for a given
census tract based on its infrastructure conditions. These conditions can be
characterized by features such as the distribution of structure built years
and the attributes of power infrastructure.

Figure 4: Structure of the conditional model, d_out=1 for the exponential loss
and d_out=2 for the weighed cross entropy loss

$F_{out}=F_{in}*scale+bias$ (1)

###  III-B Unconditional MLP Structure

The unconditional model concatenates all input features into a 1-D vector,
subsequently passing it through a series of alternating linear and non-linear
layers. These layers exhibit a monotonically decreasing number of neurons in
each layer. The structure is shown in Fig. 5.

Figure 5: Structure of the unconditional model, d_out=1 for the exponential
loss and d_out=2 for the weighed cross entropy loss

###  III-C Power Outage Probability Prediction

The prediction target is the probability of power outage where the values are
between 0 and 1. One can naturally think of using cross entropy loss. Because
of the imbalance between outage and non-outage classes, we put a large weight
on the outage class. The outputs of both models go through a Softmax layer and
the cross entropy losses are calculated by (2), where $N$ is the number of
samples in the training set, $P_{gt,0}$ and $P_{pred,0}$ are the probability
of non-outage from ground truth and model prediction, respectively. Also,
$P_{gt,1}$ and $P_{pred,1}$ are the probability of outage from ground truth
and model prediction, and $w$ is the weight for the outage class ($w=500$ in
our case).

$L=-\frac{1}{N}\sum{[P_{gt,0}\log{P_{pred,0}}+w*P_{gt,1}\log{P_{pred,1}}]}$
(2)

The second loss function employed is an exponential loss function, as depicted
in (3). Here, $P_{gt}$ and $P_{pred}$ represent the probability of power
outage in the ground truth and the model prediction, respectively. The
parameter $\beta$ serves as a scaling factor to penalize substantial
disparities between the prediction and ground truth (in our instance, $\beta$
is set to 20).

$L=\frac{1}{N}\sum{e^{|P_{gt}-P_{pred}|*\beta}}$ (3)

Both models are trained using back propagation as shown in Fig. 6.

Figure 6: Model training using back propagation.

###  III-D Evaluation Metrics

Because the outputs of Sigmoid and Softmax layers cannot reach 0, model
outputs under 0.05 are set to 0 before evaluation. We use Mean Absolute Error
(MAE) and Root Mean Squared Error (RMSE) to evaluate the model’s performance,
as shown in (4) and (5).

$MAE=\frac{1}{N}\sum{|P_{gt}-P_{pred}|}$ (4)

$RMSE=\sqrt{\frac{1}{N}\sum{|P_{gt}-P_{pred}|^{2}}}$ (5)

TABLE I: Evaluation Metrics of Different Models and Loss Functions Model | Loss Function | MAE | RMSE
---|---|---|---
unconditional | exponential | 0.01346(6.8e-3) | 0.04773(1.5e-2)
conditional | exponential | 0.0189(6.5e-3) | 0.04824(5.2e-3)
unconditional | cross entropy | 0.03243(7.1e-4) | 0.1179(2.2e-3)
conditional | cross entropy | 0.0281(1.3e-3) | 0.1131(4.4e-3)

##  IV Experiment Results

We partitioned the census tracts in the study area into three subsets: 72% for
the training set, 8% for the validation set, and 20% for the test set. Our
evaluation metrics are reported based on the test sets. The results were
averaged over three separate runs, showcasing both the mean and standard
deviations. As observed in Table I, when the exponential loss function is
used, the unconditional model exhibits lower error, whereas with the cross-
entropy loss, the conditional model demonstrates superior performance. The
variation in performance between the two models may stem from the choice of
hyper-parameters, such as the selection of the parameter $w$ in the cross-
entropy loss and $\beta$ in the exponential loss.

Predictions of both models using exponential loss for one of the census tracts
are shown in Fig. 7. As illustrated in Fig. 7, false positive is the main
source of error due to the data imbalance. A two-stage model might reduce the
false positive prediction by first classifying the outage status (outage or
non-outage) then estimating the outage customer number. Moreover, a Majority
Under-Sampling and Minority Over-Sampling strategy can be utilized to overcome
the class imbalance problem [14].

Figure 7: Predictions of both models using exponential loss for one of the census tracts. TABLE II: Ablation Study of Different Input Factors Input Factors | Exponential | Cross Entropy
---|---|---
Weather | Distance | Totals | Income | Year Built | Power Infra | MAE | RMSE | MAE | RMSE
✓ |  |  |  |  |  | 0.03547(7.3e-3) | 0.0703(4.5e-3) | 0.04901(3.5e-3) | 0.127(5.6e-3)
✓ | ✓ |  |  |  |  | 0.01916(1.0e-2) | 0.05018(1.2e-2) | 0.03652(1.6e-3) | 0.1231(3.8e-3)
✓ | ✓ | ✓ |  |  |  | 0.01015(4.3e-3) | 0.03777(8.7e-3) | 0.03662(3.1e-3) | 0.1239(5.8e-3)
✓ | ✓ | ✓ | ✓ |  |  | 0.01376(4.4e-3) | 0.04874(1.0e-2) | 0.0355(2.5e-3) | 0.1233(5.4e-3)
✓ | ✓ | ✓ | ✓ | ✓ |  | 0.01413(6.5e-3) | 0.04848(1.0e-2) | 0.03235(2.6e-3) | 0.1178(4.6e-3)
✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 0.01346(6.8e-3) | 0.04773(1.5e-2) | 0.03243(7.1e-4) | 0.1179(2.2e-3)

To investigate the impacts of different factors on the model performance, we
conducted an ablation study on the unconditional model. We gradually add more
features into the model input and see the performance variations of the model.
The mean and standard deviation of three runs are reported. The results are
shown in Table II. The inclusion of additional features leads to a reduction
in prediction errors. The variability in results for models employing the
exponential loss could potentially be attributed to noise in the household
income, year structure built, and power infrastructure distribution data.

##  V Conclusion

This paper devised both unconditional and conditional models to predict power
outage probabilities at the census tract level. To mitigate unbalanced data
issue, two distinct loss functions were implemented: exponential loss and
weighted cross entropy loss. The results showed that using the exponential
loss results in 5% less MAE in the unconditional model than the conditional
model; however, the weighted cross entropy loss showed 0.4% less MAE in the
conditional model than the unconditional model. Furthermore, the ablation
study, conducted to investigate the impact of various input features on the
model performance, showed that incorporating socio-economic factors and power
infrastructure features reduced the model’s prediction error and, therefore,
had a positive impact on the power outage probability prediction.

Future research avenues may involve further feature integration into the
model, exploring the application of LSTM to leverage historical power outage
patterns, or transitioning to a broader city or county-level analysis as a
means to partially address the data imbalance issue encountered at the finer-
grained census tract level.

## Acknowledgment

We express our gratitude to Mark Furland for generously supplying the
historical DTE outages data.

This material is based upon work supported by the Department of Energy, Solar
Energy Technologies Office (SETO) Renewables Advancing Community Energy
Resilience (RACER) program under Award Number DE-EE0010413. Any opinions,
findings, conclusions, or recommendations expressed in this material are those
of the authors and do not necessarily reflect the views of the Department of
Energy.

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