# Lax-type pairs in the theory of bivariate orthogonal polynomials

Amílcar Branquinho, Ana Foulquié-Moreno, Teresa E. Pérez,
and Miguel A. Piñar  CMUC, Department of Mathematics, University of Coimbra,
Apartado 3008, EC Santa Cruz, 3001-501 COIMBRA, Portugal<EMAIL_ADDRESS>CIDMA, Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro,
Portugal<EMAIL_ADDRESS>Instituto de Matemáticas IMAG & Departamento de
Matemática Aplicada, Facultad de Ciencias. Universidad de Granada (Spain)
<EMAIL_ADDRESS>Instituto de Matemáticas IMAG & Departamento de Matemática
Aplicada, Facultad de Ciencias. Universidad de Granada (Spain<EMAIL_ADDRESS>
###### Abstract.

Sequences of bivariate orthogonal polynomials written as vector polynomials of
increasing size satisfy a couple of three term relations with matrix
coefficients. In this work, introducing a time-dependent parameter, we analyse
a Lax-type pair system for the coefficients of the three term relations. We
also deduce several characterizations relating the Lax-type pair, the shape of
the weight, Stieltjes function, moments, a differential equation for the
weight, and the bidimensional Toda-type systems.

###### Key words and phrases:

Two variable orthogonal polynomials, $2$D Toda lattice, Block Lax pairs

###### 2020 Mathematics Subject Classification:

Primary: 42C05; 33C50; 35Q53

AB acknowledges Centro de Matemática da Universidade de Coimbra
(CMUC)—UIDB/00324/2020 (funded by the Portuguese Government through
FCT/MCTES).
AFM acknowledge CIDMA—Center for Research & Development in Mathematics and
Applications is supported through the Portuguese Foundation for Science and
Technology (FCT—Fundação para a Ciência e a Tecnologia), references
UIDB/04106/2020 and UIDP/04106/2020.
TEP and MAP thanks Grant FQM-246-UGR20 funded by Consejería de Universidad,
Investigación e Innovación de la Junta de Andalucía and FEDER, Una manera de
Hacer Europa; IMAG-María de Maeztu grant CEX2020-001105-M, and Research Group
Goya FQM-384

##  1\. Introduction

Several areas of Classical Analysis have profound connections with the theory
of orthogonal polynomials, such as moment problems, spectral theory of Jacobi
matrices, and random matrices.

One of the more recent connection is with the integrable systems that starts
in the 1970’s with the work by Gardner, Greene, Kruskal and Miura in [11, 12],
with a method for the exact solution of the initial-value problem for the KdV
equation, which is now referred to as the inverse scattering transform [1]. In
[15], Lax put the inverse scattering method for solving the KdV equation into
a more general framework, which subsequently paved the way to generalizations
of the technique, as a method for solving other partial differential
equations.

In fact, he considered, associated with the KdV equation, two time dependent
operators $\mathcal{L}$ and $\mathcal{M}$, where $\mathcal{L}$ is the operator
of the spectral problem and $\mathcal{M}$ is the operator governing the
associated time evolution of the eigenfunctions

$\displaystyle\mathcal{L}v=\lambda v,$
$\displaystyle\dfrac{\operatorname{d}}{\operatorname{d}t}v=\mathcal{M}v,$

and hence

$\displaystyle\dfrac{\operatorname{d}}{\operatorname{d}t}\mathcal{L}=\mathcal{M}\,\mathcal{L}-\mathcal{L}\,\mathcal{M},$
if, and only if,
$\displaystyle\dfrac{\operatorname{d}}{\operatorname{d}t}\lambda=0.$

The relation between Toda lattices and univariate orthogonality has been
analysed by several authors, for an introduction and first properties, [4, 14,
17, 18], among many others, can be consulted. In the multivariate case, some
papers studied the case when several continuous time dependent variables are
considered, see [6, 7], and in [5] multidimensional analogues of continuous
and discrete-time Toda lattices with two or more space coordinates relating
them with multiple orthogonal polynomials are studied.

The $2$D Toda lattices with only a time dependent variable can be seen in [8].
There the authors depart from the knowledge of the weight representation and
derive some Toda like lattices and at the end a Lax type representation for
them. In this paper we apply the method of moments, presented in [3] for the
univariate Toda lattice, to a theory so far general of $2$D Toda lattice.

Usually, orthogonal polynomials in two variables are written as vector
polynomials of increasing size, and they satisfy three term relations with
matrix coefficients (see [10]). We consider orthogonality with respect to a
bivariate weight function and introduce a time-dependent parameter such that
we work with an evolution weight function. Here we depart from the two
tridiagonal block Jacobi operators and present the correspondence between
dynamics of the $2$D Toda equations for the block-coefficients of the
operators and the Stieltjes function associated with the block Jacobi
operators. This gives a method to solve an inverse problem, i.e., we can get
an integral representation for the coefficients of the block Jacobi matrices
that satisfies a $2$D Toda equations in terms of a weight function completely
determined by the data.

The work is organized in four sections. After this introduction, in Section 2
we briefly present the theory and basic facts of bivariate orthogonal
polynomials that we will need in the sequel. In Section 3 we give, in Theorem
3.1, an interpretation of the $2$D Toda lattice in terms of the theory of
bivariate orthogonal polynomials. This is in fact the main results of the
paper, where we characterize the $2$D Toda lattice in terms of the moments and
the Stieltjes function associated with the $2$D Toda lattice. There we also
give a representation for the weight that governs the $2$D Toda lattice, and
deduce several characterizations relating the Lax-type pair, the moments, a
differential equation for the weight, and the bidimensional Toda-type system.
Finally, in Section 4 we end the work by establishing a Lax type theorem and
proving the isospectrality of the associated block Jacobi matrices.

##  2\. Preliminary results

For each $n\geqslant 0$, let $\Pi_{n}$ denote the linear space of bivariate
polynomials of total degree not greater than $n$ (cf. [10]). We consider
$\Pi=\bigcup_{n\geqslant 0}\Pi_{n}$ the linear space of all bivariate
polynomials with real coefficients.

We say that $p(x,y)\in\Pi_{n}$ is _monomial_ if there is only one term of
higher degree, i.e., there exists $0\leqslant k\leqslant n$ such that

$\displaystyle
p(x,y)=a_{n-k,k}x^{n-k}y^{k}+\sum_{m=0}^{n-1}\sum_{i=0}^{m}a_{m-i,i}x^{m-i}y^{i},$

with $a_{n-k,k}\neq 0$. Moreover, if $a_{n-k,k}=1$, we say that $p(x,y)$ is
_monic_.

Let us denote by $\mathcal{M}_{h\times k}(\mathbb{R})$ the linear space of
matrices of size $h\times k$ with real entries and by $\mathcal{M}_{h\times
k}(\Pi)$ the linear space of $h\times k$ matrices with polynomials in two
variables entries. The _degree of a matrix polynomial_ is defined as the
maximum of the degrees of its polynomial entries.

Given a matrix $\operatorname{M}\in\mathcal{M}_{h\times k}$, we denote by
$\operatorname{M}^{\top}$ its transpose. If $h=k$, we will denote
$\mathcal{M}_{h\times h}\equiv\mathcal{M}_{h}$. In particular,
$\operatorname{I}_{h}$ denotes the identity matrix of size $h$. We say that
$\operatorname{M}\in\mathcal{M}_{h}(\mathbb{R})$ is _non-singular_ if
$\det\operatorname{M}\neq 0$.

###  2.1. Vector notation

For each $n\geqslant 0$, let $\mathbb{X}_{n}$ denote the column vector

$\displaystyle\mathbb{X}_{n}=\begin{bmatrix}x^{n}&x^{n-1}y&\cdots&xy^{n-1}&y^{n}\end{bmatrix}^{\top},$

of size $(n+1)\times 1$. Then $\big{\\{}\mathbb{X}_{n}\big{\\}}_{n\geqslant
0}$ is called the _canonical basis_ of $\Pi$ and every polynomial
$\operatorname{P}\in\Pi$ of degree $n$ can be represented as

$\displaystyle\operatorname{P}(x,y)=\sum_{k=0}^{n}\operatorname{C}_{k}^{\top}\,\mathbb{X}_{k},$

where $\operatorname{C}_{k}$ is a $(k+1)\times 1$ vector of constants. We
continue, following the notation in [10], for $n\geqslant 0$, denoting
$\operatorname{L}_{n,1}$ and $\operatorname{L}_{n,2}$, as the matrices of size
$(n+1)\times(n+2)$, such that

$\displaystyle\operatorname{L}_{n,1}\,\mathbb{X}_{n+1}=x\,\mathbb{X}_{n}$ and
$\displaystyle\operatorname{L}_{n,2}\,\mathbb{X}_{n+1}=y\,\mathbb{X}_{n},$

where, $\operatorname{L}_{n,1}$ and $\operatorname{L}_{n,2}$ represent the so-
called shift operators associated with the multiplication of $\mathbb{X}_{n}$
by the variables $x$ and $y$, respectively. Therefore,

$\displaystyle\operatorname{L}_{n,1}=\left[\begin{array}[]{ccc|c}1&{}\hfil&{}\hfil&0\\\\[-3.55658pt]
{}\hfil&\ddots&{}\hfil&\vdots\\\ {}\hfil&{}\hfil&1&0\end{array}\right],$
$\displaystyle\operatorname{L}_{n,2}=\left[\begin{array}[]{c|ccc}0&1&{}\hfil&{}\hfil\\\\[-3.55658pt]
\vdots&{}\hfil&\ddots&{}\hfil\\\ 0&{}\hfil&{}\hfil&1\end{array}\right].$ (2.7)

Observe that, $\operatorname{rank}\operatorname{L}_{n,i}=n+1$,
$\operatorname{L}_{n,i}\operatorname{L}_{n,i}^{\top}=\operatorname{I}_{n+1}$,
for $i=1,2$, and $\operatorname{L}_{n,1}\operatorname{L}_{n+1,2}$
$=\operatorname{L}_{n,2}\operatorname{L}_{n+1,1}$.

Let $\big{\\{}\operatorname{P}_{n,m}(x,y):0\leqslant m\leqslant n,n\geqslant
0\big{\\}}$ denote a basis of $\Pi$ such that, for a fixed $n\geqslant 0$,
$\deg\operatorname{P}_{n,m}(x,y)=n$, and the set
$\big{\\{}\operatorname{P}_{n,m}(x,y):0\leqslant m\leqslant n\big{\\}}$
contains $n+1$ linearly independent polynomials of total degree exactly $n$.
We can write the vector of polynomials

$\displaystyle\mathbb{P}_{n}=\begin{bmatrix}\operatorname{P}_{n,0}(x,y)&\operatorname{P}_{n,1}(x,y)&\cdots&\operatorname{P}_{n,n}(x,y)\end{bmatrix}^{\top}.$

The sequence of polynomial vectors of increasing size
$\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ is called a _polynomial
system_ (PS), and it is a basis of $\Pi$. Moreover, for each $n\geqslant 0$,
the vector $\mathbb{P}_{n}$ can be written as

$\displaystyle\mathbb{P}_{n}=\operatorname{G}_{n}^{n}\,\mathbb{X}_{n}+\operatorname{G}^{n}_{n-1}\,\mathbb{X}_{n-1}+\cdots+\operatorname{G}^{n}_{0}\,\mathbb{X}_{0},$

where $\operatorname{G}_{n}^{n}$ is a $n+1$ non-singular matrix of constants,
called _the matrix leading coefficient_ of $\mathbb{P}_{n}$, and
$\operatorname{G}^{n}_{m}$ are $(n+1)\times(m+1)$ constant matrices. When
$\operatorname{G}_{n}^{n}$ is the identity matrix, i.e.,
$\operatorname{G}_{n}^{n}=\operatorname{I}_{n+1}$, for every $n\geqslant 0$,
then we say that $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ is a _monic
PS_. Observe that the system of polynomials given by
$\widehat{\mathbb{P}}_{n}=(\operatorname{G}_{n}^{n})^{-1}\,\mathbb{P}_{n}$,
for $n\geqslant 0$, is a monic polynomial system, in the sense that

$\displaystyle\widehat{\mathbb{P}}_{n}=\mathbb{X}_{n}+\widehat{\operatorname{G}}^{n}_{n-1}\,\mathbb{X}_{n-1}+\cdots+\widehat{\operatorname{G}}^{n}_{0}\,\mathbb{X}_{0},$

where
$\widehat{\operatorname{G}}^{n}_{m}=(\operatorname{G}_{n}^{n})^{-1}\,\operatorname{G}^{n}_{m}$,
for $0\leqslant m\leqslant n-1$.

###  2.2. Orthogonal polynomial systems (OPS)

Let $\operatorname{d}\mu(x,y)$ be a normalized measure defined on a region
$\Omega\subset\mathbb{R}^{2}$ and we assume that it is positive definite, in
the way that

$\displaystyle\int_{\Omega}p^{2}(x,y)\,\operatorname{d}\mu(x,y)>0,$
$\displaystyle p\in\Pi,$ $\displaystyle p\not\equiv 0.$

Moreover, we suppose that the moments

$\displaystyle\omega_{h,k}=\int_{\Omega}x^{h}\,y^{k}\,\operatorname{d}\mu(x,y)<+\infty,$
$\displaystyle h,k\geqslant 0,$ (2.8)

exist, and

$\displaystyle\omega_{0,0}=\int_{\Omega}\operatorname{d}\mu(x,y)=1.$

As usual, we define the inner product

$\displaystyle\langle
p,q\rangle=\int_{\Omega}p(x,y)\,q(x,y)\,\operatorname{d}\mu(x,y),$
$\displaystyle p,q\in\Pi,$ with $\displaystyle\langle 1,1\rangle=1.$

We will see how the inner product acts over polynomial matrices. Let
$\operatorname{A}=\begin{bmatrix}a_{i,j}(x,y)\end{bmatrix}_{i,j=1}^{h,k}$ and
$\operatorname{B}=\begin{bmatrix}b_{i,j}(x,y)\end{bmatrix}_{i,j=1}^{k,l}$ be
two polynomial matrices, i.e., $a_{i,j}(x,y),b_{i,j}(x,y)\in\Pi$. The action
of the above inner product over polynomial matrices is defined as the $h\times
l$ matrix (cf. [10]),

$\displaystyle\langle\operatorname{A},\operatorname{B}\rangle=\int_{\Omega}\operatorname{A}(x,y)\,\operatorname{B}(x,y)\,\operatorname{d}\mu(x,y)=\begin{bmatrix}\displaystyle\int_{\Omega}c_{i,j}(x,y)\operatorname{d}\mu(x,y)\end{bmatrix}_{i,j=1}^{h,l},$

where
$\displaystyle\operatorname{C}=\operatorname{A}\cdot\operatorname{B}=\begin{bmatrix}c_{i,j}(x,y)\end{bmatrix}_{i,j=1}^{h,l}$.
We say that $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ is an
_orthogonal polynomial system (OPS) with respect to
$\langle\cdot,\cdot\rangle$_ if

$\displaystyle\langle\mathbb{P}_{n},\mathbb{P}^{\top}_{m}\rangle=\begin{cases}\mathtt{0}_{(n+1)\times(m+1)},&n\neq
m,\\\ \operatorname{H}_{n},&n=m,\end{cases}$ (2.9)

where $\operatorname{H}_{n}$ is a symmetric and positive-definite matrix of
size $n+1$, and $\mathtt{0}_{(n+1)\times(m+1)}$, or $\mathtt{0}$ for short, is
the zero matrix of adequate size. When $\operatorname{H}_{n}$ is diagonal,
$n\in\mathbb{N}$, we say that $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant
0}$ is a _mutually orthogonal polynomial system_. Moreover, there exists an
_orthonormal PS_ satisfying (2.9) with
$\operatorname{H}_{n}=\operatorname{I}_{n+1}$. In addition, there exists a
unique monic orthogonal polynomial system associated to
$\operatorname{d}\mu(x,y)$.

###  2.3. Three term relations

Orthogonal polynomials in two variables satisfy, in each variable, a three
term relation. These three term relations are written in a vector form and
have matrix coefficients.

######  Proposition 2.1 (cf. [10]).

Let $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ be a monic OPS
associated to an inner product $\langle\cdot,\cdot\rangle$. For $n\geqslant
0$, there exist constant matrices $\operatorname{D}_{n,i}$ and
$\operatorname{C}_{n,i}$ of sizes respectively given by $(n+1)\times(n+1)$ and
$(n+1)\times n$, $i=1,2$, such that

$\displaystyle\begin{cases}x\,\mathbb{P}_{n}=\operatorname{L}_{n,1}\,\mathbb{P}_{n+1}+\operatorname{D}_{n,1}\,\mathbb{P}_{n}+\operatorname{C}_{n,1}\,\mathbb{P}_{n-1},\\\
y\,\mathbb{P}_{n}=\operatorname{L}_{n,2}\,\mathbb{P}_{n+1}+\operatorname{D}_{n,2}\,\mathbb{P}_{n}+\operatorname{C}_{n,2}\,\mathbb{P}_{n-1},\end{cases}$
(2.10)

where $\mathbb{P}_{-1}=0$, $\operatorname{C}_{-1,i}=0$, and

$\displaystyle\operatorname{D}_{n,1}\,\operatorname{H}_{n}=\langle
x\,\mathbb{P}_{n},\,\mathbb{P}_{n}^{\top}\rangle,$
$\displaystyle\operatorname{D}_{n,2}\,\operatorname{H}_{n}=\langle
y\,\mathbb{P}_{n},\,\mathbb{P}_{n}^{\top}\rangle,$
$\displaystyle\operatorname{C}_{n,1}\,\operatorname{H}_{n-1}=\operatorname{H}_{n}\,\operatorname{L}_{n-1,1}^{\top},$
$\displaystyle\operatorname{C}_{n,2}\,\operatorname{H}_{n-1}=\operatorname{H}_{n}\,\operatorname{L}_{n-1,2}^{\top}.$

Moreover, the rank conditions

$\displaystyle\operatorname{rank}\operatorname{C}_{n,i}=n,\quad i=1,2,$
$\displaystyle\operatorname{rank}\begin{bmatrix}\operatorname{C}_{n,1}&\operatorname{C}_{n,2}\end{bmatrix}=n+1,$

hold.

On the other hand, we can add the relations in (2.10), to get

$\displaystyle(x+y)\,\mathbb{P}_{n}=\operatorname{L}_{n}\,\mathbb{P}_{n+1}+\operatorname{D}_{n}\,\mathbb{P}_{n}+\operatorname{C}_{n}\,\mathbb{P}_{n-1},$
(2.11)

where

$\displaystyle\operatorname{L}_{n}$
$\displaystyle=\operatorname{L}_{n,1}+\operatorname{L}_{n,2}\,\in\,\mathcal{M}_{(n+1)\times(n+2)}(\mathbb{R}),$
$\displaystyle\operatorname{D}_{n}$
$\displaystyle=\operatorname{D}_{n,1}+\operatorname{D}_{n,2}\,\in\,\mathcal{M}_{(n+1)\times(n+1)}(\mathbb{R}),$
$\displaystyle\operatorname{C}_{n}$
$\displaystyle=\operatorname{C}_{n,1}+\operatorname{C}_{n,2}\,\in\,\mathcal{M}_{(n+1)\times
n}(\mathbb{R}).$

Here we pointed out that, taking $n=0$ in (2.10), and multiplying by
$\mathbb{P}_{0}=1$, we get

$\displaystyle\langle x\mathbb{P}_{0},\mathbb{P}_{0}\rangle$
$\displaystyle=\operatorname{L}_{0,1}\langle\mathbb{P}_{1},\mathbb{P}_{0}\rangle+\operatorname{D}_{0,1}\langle\mathbb{P}_{0},\mathbb{P}_{0}\rangle,$
$\displaystyle\langle y\mathbb{P}_{0},\mathbb{P}_{0}\rangle$
$\displaystyle=\operatorname{L}_{0,2}\langle\mathbb{P}_{1},\mathbb{P}_{0}\rangle+\operatorname{D}_{0,2}\langle\mathbb{P}_{0},\mathbb{P}_{0}\rangle,$

and then

$\displaystyle\omega_{1,0}=\operatorname{D}_{0,1},$
$\displaystyle\omega_{0,1}=\operatorname{D}_{0,2}.$

since $\langle\mathbb{P}_{0},\mathbb{P}_{0}\rangle=\langle
1,1\rangle=\omega_{0,0}=1$.

For the orthonormal polynomials
$\big{\\{}\widetilde{\mathbb{P}}_{n}\big{\\}}_{n\geqslant 0}$, such that
$\operatorname{H}_{n}=\operatorname{I}_{n+1}$, the three term relations (2.10)
take a simpler form, as we can see in the next result.

######  Proposition 2.2 ([10]).

For $n\geqslant 0$, there exist real matrices $\operatorname{A}_{n,i}$,
$\operatorname{B}_{n,i}$ of respective sizes $(n+1)\times(n+2)$ and
$(n+1)\times(n+1)$ such that

$\displaystyle\begin{aligned}
&x\,\widetilde{\mathbb{P}}_{n}=\operatorname{A}_{n,1}\,\widetilde{\mathbb{P}}_{n+1}+\operatorname{B}_{n,1}\,\widetilde{\mathbb{P}}_{n}+\operatorname{A}^{\top}_{n-1,1}\,\widetilde{\mathbb{P}}_{n-1},\\\
&y\,\widetilde{\mathbb{P}}_{n}=\operatorname{A}_{n,2}\,\widetilde{\mathbb{P}}_{n+1}+\operatorname{B}_{n,2}\,\widetilde{\mathbb{P}}_{n}+\operatorname{A}^{\top}_{n-1,2}\,\widetilde{\mathbb{P}}_{n-1},\end{aligned}$
(2.12)

where we define $\widetilde{\mathbb{P}}_{-1}=0$ and
$\operatorname{A}_{-1,i}=0$. Moreover, each $\operatorname{B}_{n,i}$ is
symmetric.

Following [10, p. 64], given an OPS
$\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ we can construct an
orthonormal PS in the form

$\displaystyle\widetilde{\mathbb{P}}_{n}=\operatorname{H}_{n}^{-1/2}\,\mathbb{P}_{n},$

where
$\operatorname{H}_{n}=\langle\mathbb{P}_{n},\mathbb{P}^{\top}_{n}\rangle$, and
$\operatorname{H}_{n}^{-1/2}=(\operatorname{H}_{n}^{1/2})^{-1}$ is the inverse
of the positive square root of $\operatorname{H}_{n}$, the unique positive-
definite matrix satisfying
$\operatorname{H}_{n}^{1/2}\,\operatorname{H}_{n}^{1/2}=\operatorname{H}_{n}$
(cf. [13]). In such a case, we get

$\displaystyle\operatorname{A}_{n,i}=\operatorname{H}_{n}^{-1/2}\,\operatorname{L}_{n,i}\,\operatorname{H}_{n+1}^{1/2},$
$\displaystyle\operatorname{B}_{n,i}=\operatorname{H}_{n}^{-1/2}\,\operatorname{D}_{n,i}\,\operatorname{H}_{n}^{1/2}.$

###  2.4. Block Jacobi matrices

As in the one variable case, we can consider in the two variables one, two
infinite tridiagonal block matrices $\boldsymbol{\operatorname{J}}_{i}$,
$i=1,2$, defined as follows

$\displaystyle\boldsymbol{\operatorname{J}}_{i}=\begin{bmatrix}\operatorname{D}_{0,i}&\operatorname{L}_{0,i}&&&\\\
\operatorname{C}_{1,i}&\operatorname{D}_{1,i}&\operatorname{L}_{1,i}&&\\\
&\operatorname{C}_{2,i}&\operatorname{D}_{2,i}&\operatorname{L}_{2,i}&\\\
&&\ddots&\ddots&\ddots\end{bmatrix},$ (2.13)

where the matrices in the main diagonal $\operatorname{D}_{n,i}$ are square
matrices of increasing size $n+1$, the matrices in the subdiagonal,
$\operatorname{C}_{n,i}$, are $(n+1)\times n$ matrices, and in the
superdiagonal, the matrices $\operatorname{L}_{n,i}$ are $n\times(n+1)$
matrices, and the other elements are zero matrices of appropriate size.

Defining the infinite column vector

$\displaystyle\boldsymbol{\operatorname{P}}=\begin{bmatrix}\mathbb{P}^{\top}_{0}&\mathbb{P}^{\top}_{1}&\cdots&\mathbb{P}^{\top}_{n}&\cdots\end{bmatrix}^{\top},$

the three term relations (2.10) can be written as

$\displaystyle
x\,\boldsymbol{\operatorname{P}}=\boldsymbol{\operatorname{J}}_{1}\,\boldsymbol{\operatorname{P}},$
$\displaystyle
y\,\boldsymbol{\operatorname{P}}=\boldsymbol{\operatorname{J}}_{2}\,\boldsymbol{\operatorname{P}}.$

Directly from the three term relation we have a formal result above the
commutativity of the matrix entries of both block matrices.

######  Lemma 2.3.

For $n\geqslant 0$, we get

$\displaystyle\begin{aligned}
\phantom{ola}&\operatorname{C}_{n,1}\operatorname{C}_{n-1,2}=\operatorname{C}_{n,2}\operatorname{C}_{n-1,1},\\\
\phantom{ola}&\operatorname{D}_{n,1}\,\operatorname{C}_{n,2}+\operatorname{C}_{n,1}\,\operatorname{D}_{n-1,2}=\operatorname{D}_{n,2}\,\operatorname{C}_{n,1}+\operatorname{C}_{n,2}\,\operatorname{D}_{n-1,1},\\\
\phantom{ola}&\operatorname{L}_{n,1}\operatorname{C}_{n+1,2}+\operatorname{D}_{n,1}\operatorname{D}_{n,2}+\operatorname{C}_{n,1}\operatorname{L}_{n-1,2}\\\
&\phantom{olaolaolaola}=\operatorname{L}_{n,2}\operatorname{C}_{n+1,1}+\operatorname{D}_{n,2}\operatorname{D}_{n,1}+\operatorname{C}_{n,2}\operatorname{L}_{n-1,1},\\\
\phantom{ola}&\operatorname{L}_{n,1}\operatorname{D}_{n+1,2}+\operatorname{D}_{n,1}\operatorname{L}_{n,2}=\operatorname{L}_{n,2}\operatorname{D}_{n+1,1}+\operatorname{D}_{n,2}\operatorname{L}_{n,1},\end{aligned}$
(2.14)

where we assume $\operatorname{C}_{0,1}=\operatorname{C}_{0,2}=0$.

###### Proof.

Since $y\,x\,\mathbb{P}_{n}=x\,y\,\mathbb{P}_{n}$, from (2.10) we get

$\operatorname{L}_{n,1}\big{(}\operatorname{L}_{n+1,2}\,\mathbb{P}_{n+2}+\operatorname{D}_{n+1,2}\,\mathbb{P}_{n+1}+\operatorname{C}_{n+1,2}\,\mathbb{P}_{n})\\\
+\operatorname{D}_{n,1}\big{(}\operatorname{L}_{n,2}\,\mathbb{P}_{n+1}+\operatorname{D}_{n,2}\,\mathbb{P}_{n}+\operatorname{C}_{n,2}\,\mathbb{P}_{n-1}\big{)}\\\
+\operatorname{C}_{n,1}\big{(}\operatorname{L}_{n-1,2}\,\mathbb{P}_{n}+\operatorname{D}_{n-1,2}\,\mathbb{P}_{n-1}+\operatorname{C}_{n-1,2}\,\mathbb{P}_{n-2}\big{)}\\\
=\operatorname{L}_{n,2}\big{(}\operatorname{L}_{n+1,1}\,\mathbb{P}_{n+2}+\operatorname{D}_{n+1,1}\,\mathbb{P}_{n+1}+\operatorname{C}_{n+1,1}\,\mathbb{P}_{n}\big{)}\\\
+\operatorname{D}_{n,2}\big{(}\operatorname{L}_{n,1}\,\mathbb{P}_{n+1}+\operatorname{D}_{n,1}\,\mathbb{P}_{n}+\operatorname{C}_{n,1}\,\mathbb{P}_{n-1}\big{)}\\\
+\operatorname{C}_{n,2}\big{(}\operatorname{L}_{n-1,1}\,\mathbb{P}_{n}+\operatorname{D}_{n-1,1}\,\mathbb{P}_{n-1}+\operatorname{C}_{n-1,1}\,\mathbb{P}_{n-2}\big{)}.$

Taking $\langle y\,x\,\mathbb{P}_{n},\mathbb{P}_{j}^{\top}\rangle=\langle
x\,y\,\mathbb{P}_{n},\mathbb{P}_{j}^{\top}\rangle$, successively for
$j=n-2$,$n-1$,$n$, $n+1$,$n+2$ we get the desired result. ∎

As it is shown in [10, pp. 83 and seq.], above conditions are not enough to
assure the commutativity of $\boldsymbol{\operatorname{J}}_{1}$ and
$\boldsymbol{\operatorname{J}}_{2}$ since they are operators. The results
about commutativity are showed for the Jacobi matrix for orthonormal
polynomial systems. We consider the Jacobi block matrix based on the three
term relations for the orthonormal polynomials (2.12):

$\displaystyle\boldsymbol{\operatorname{L}}_{i}=\begin{bmatrix}\operatorname{B}_{0,i}&\operatorname{A}_{0,i}&&\\\
\operatorname{A}^{\top}_{0,i}&\operatorname{B}_{1,i}&\operatorname{A}_{1,i}&\\\
&\operatorname{A}^{\top}_{1,i}&\operatorname{B}_{2,i}&\operatorname{A}_{2,i}\\\
&&\ddots&\ddots&\ddots\end{bmatrix},$ $\displaystyle i=1,2,$ (2.15)

where the matrices $\operatorname{A}_{n,i}$ and $\operatorname{B}_{n,i}$ are
given in the three term relations, and the other elements are zero matrices of
appropriate size. Then the three term relations (2.12) can be written as

$\displaystyle
x\,\widetilde{\boldsymbol{\operatorname{P}}}=\boldsymbol{\operatorname{L}}_{1}\,\widetilde{\boldsymbol{\operatorname{P}}},$
$\displaystyle
y\,\widetilde{\boldsymbol{\operatorname{P}}}=\boldsymbol{\operatorname{L}}_{2}\,\widetilde{\boldsymbol{\operatorname{P}}},$

where
$\widetilde{\boldsymbol{\operatorname{P}}}=\begin{bmatrix}\widetilde{\mathbb{P}}^{\top}_{0}&\widetilde{\mathbb{P}}^{\top}_{1}&\cdots&\widetilde{\mathbb{P}}^{\top}_{n}&\cdots\end{bmatrix}^{\top}$
is the vector of the orthonormal polynomials.

The matrices $\boldsymbol{\operatorname{L}}_{1}$ and
$\boldsymbol{\operatorname{L}}_{2}$ can be considered as linear operators
which act via matrix multiplication on $\ell^{2}$, where the domain of the
operator consists of all sequences in $\ell^{2}$ for which matrix
multiplication yields sequences in $\ell^{2}$. As linear operators, the
matrices $\boldsymbol{\operatorname{L}}_{1}$ and
$\boldsymbol{\operatorname{L}}_{2}$, does not need to be bounded, however,
according to [10, pp. 83 and seq.], a sufficient condition for the boundedness
of $\boldsymbol{\operatorname{L}}_{1}$ and $\boldsymbol{\operatorname{L}}_{2}$
is the compactness of the support $\Omega$ of de measure $\mu$ having the
sequence $\left\\{\widetilde{\mathbb{P}}_{n}\right\\}_{n\geqslant 0}$ as
orthonormal polynomials.

Now, defining

$\displaystyle\boldsymbol{\operatorname{H}}=\begin{bmatrix}\operatorname{H}_{0}&&&\\\
&\operatorname{H}_{1}&&\\\ &&\ddots\end{bmatrix},$
$\displaystyle\boldsymbol{\operatorname{H}}^{1/2}=\begin{bmatrix}\operatorname{H}_{0}^{1/2}&&&\\\
&\operatorname{H}_{1}^{1/2}&&\\\ &&\ddots\end{bmatrix},$

and $\boldsymbol{\operatorname{H}}^{-1}$,
$\boldsymbol{\operatorname{H}}^{-1/2}$, their respective inverse matrices, we
have,

$\displaystyle\boldsymbol{\operatorname{J}}_{i}=\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{i}\,\boldsymbol{\operatorname{H}}^{-1/2},$
$\displaystyle\boldsymbol{\operatorname{L}}_{i}=\boldsymbol{\operatorname{H}}^{-1/2}\,\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{H}}^{1/2},$
$\displaystyle i=1,2.$

Following Lemma 3.4.4 in [10, p. 85], and supposing that
$\boldsymbol{\operatorname{L}}_{i}$ are bounded, then they are self-adjoint
and commute. In this way,

$\displaystyle\boldsymbol{\operatorname{J}}_{1}\,\boldsymbol{\operatorname{J}}_{2}$
$\displaystyle=\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{1}\,\boldsymbol{\operatorname{H}}^{-1/2}\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{2}\,\boldsymbol{\operatorname{H}}^{-1/2}=\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{1}\,\boldsymbol{\operatorname{L}}_{2}\,\boldsymbol{\operatorname{H}}^{-1/2}$
$\displaystyle=\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{2}\,\boldsymbol{\operatorname{L}}_{1}\,\boldsymbol{\operatorname{H}}^{-1/2}=\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{2}\,\boldsymbol{\operatorname{H}}^{-1/2}\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{1}\,\boldsymbol{\operatorname{H}}^{-1/2}=\boldsymbol{\operatorname{J}}_{2}\,\boldsymbol{\operatorname{J}}_{1}.$

We can summarize this in the following result.

######  Lemma 2.4.

If
$\boldsymbol{\operatorname{L}}_{i}=\boldsymbol{\operatorname{H}}^{-1/2}\,\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{H}}^{1/2}$,
$i=1,2$, are bounded matrices, then

$\displaystyle\boldsymbol{\operatorname{J}}_{1}\,\boldsymbol{\operatorname{J}}_{2}=\boldsymbol{\operatorname{J}}_{2}\,\boldsymbol{\operatorname{J}}_{1}.$

In general, for the bounded case we have,

$\displaystyle\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}=\boldsymbol{\operatorname{J}}_{2}^{k}\,\boldsymbol{\operatorname{J}}_{1}^{h},$
$\displaystyle h,k\geqslant 0,$

and the product
$\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}$
preserves the block structure described in (2.13), but not the zero block
matrices. In particular, we write

$\displaystyle\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}=\begin{bmatrix}\operatorname{D}^{(h,k)}_{0,0}&\operatorname{L}^{(h,k)}_{0,1}&(*)&\cdots\\\
\operatorname{C}^{(h,k)}_{1,0}&\operatorname{D}^{(h,k)}_{1,1}&\operatorname{L}^{(h,k)}_{1,2}&\ddots\\\
(*)&\operatorname{C}^{(h,k)}_{2,1}&\operatorname{D}^{(h,k)}_{2,2}&\ddots\\\
\vdots&\ddots&\ddots&\ddots\end{bmatrix},$

where the $(*)$ denotes block matrices of adequate size not necessarily zero.
For $h,k,n\geqslant 0$, the respective size of the involved matrices are
$(n+1)\times(n+1)$ for $\operatorname{D}^{(h,k)}_{n,n}$, $(n+2)\times(n+1)$
for $\operatorname{C}^{(h,k)}_{n+1,n}$, and $(n+1)\times(n+2)$ for
$\operatorname{L}^{(h,k)}_{n,n+1}$.

Observe that

$\displaystyle\operatorname{D}^{(1,0)}_{n,n}=\operatorname{D}_{n,1},$
$\displaystyle\operatorname{C}^{(1,0)}_{n,n-1}=\operatorname{C}_{n,1},$
$\displaystyle\operatorname{L}^{(1,0)}_{n-1,n}=\operatorname{L}_{n-1,1},$
$\displaystyle\operatorname{D}^{(0,1)}_{n,n}=\operatorname{D}_{n,2},$
$\displaystyle\operatorname{C}^{(0,1)}_{n,n-1}=\operatorname{C}_{n,2},$
$\displaystyle\operatorname{L}^{(0,1)}_{n-1,n}=\operatorname{L}_{n-1,2}.$

Let $\ell_{0}=\begin{bmatrix}1&0&0&\cdots\end{bmatrix}^{\top}$ be the zero
column vector except for the first element. Then, it is clear that

$\displaystyle\operatorname{D}^{(h,k)}_{0,0}=\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{0},$

i.e., we can access to the $(0,0)$ element of the matrix
$\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}$.

Next result shows that all the moments are determined by the powers of the
Jacobi matrices.

######  Proposition 2.5.

For $h,k\geqslant 0$,

$\displaystyle\omega_{h,k}=\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{0}.$
(2.16)

###### Proof.

Observe that, for all $h,k\geqslant 0$, we have

$\displaystyle
x^{h}y^{k}\,\boldsymbol{\operatorname{P}}=\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\boldsymbol{\operatorname{P}},$

and then, the first element in both sides are equal,

$\displaystyle
x^{h}y^{k}\,\mathbb{P}_{0}=\operatorname{D}_{0,0}^{(h,k)}\,\mathbb{P}_{0}+\operatorname{L}^{(h,k)}_{0,1}\,\mathbb{P}_{1}+(*)\,\mathbb{P}_{2}+\cdots$

Integrating by means of the measure $\operatorname{d}\mu(x,y)$, using the
orthogonality, and the fact that $\mathbb{P}_{0}=1$, we arrive to
$\displaystyle\displaystyle\omega_{h,k}=\operatorname{D}_{0,0}^{(h,k)}\omega_{0,0}=\operatorname{D}_{0,0}^{(h,k)}$.
∎

###  2.5. Stieltjes function in two variables

From now on, we suppose that the positive definite measure is defined on a
compact domain $\Omega\subset\mathbb{R}^{2}$ and is given in terms of a weight
function in the form:

$\displaystyle\operatorname{d}\mu(x,y)=\omega(x,y)\operatorname{d}x\operatorname{d}y.$

Suppose that all the moments in (2.8) exist. Following [2], we define the
Stieltjes function in two variables as a double formal power series.

######  Definition 2.6.

The _Stieltjes function_ associated with a weight function $\omega(x,y)$ (or,
equivalently, the operators $\boldsymbol{\operatorname{J}}_{1}$ and
$\boldsymbol{\operatorname{J}}_{2}$) is defined as

$\displaystyle\operatorname{S}(z_{1},z_{2})=\int_{\Omega}\frac{\omega(x,y)}{(z_{1}-x)\,(z_{2}-y)}\operatorname{d}x\operatorname{d}y.$
(2.17)

Observe that, substituting the formal series (cf. [9, p. 134])

$\displaystyle\frac{1}{(z_{1}-x)\,(z_{2}-y)}=\sum_{h,k\geqslant
0}\,\frac{x^{h}\,y^{k}}{z_{1}^{h+1}z_{2}^{k+1}},$

defined in
$\big{\\{}(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|>\|{|}\boldsymbol{\operatorname{J}}_{1}\|{|}\mbox{
and }\ |z_{2}|>\|{|}\boldsymbol{\operatorname{J}}_{2}\|{|}\big{\\}}$, where by
$\|{|}\boldsymbol{\operatorname{J}}_{i}\|{|}$ we mean the operator norm of
$\boldsymbol{\operatorname{J}}_{i}$, $i=1,2$, we get

$\displaystyle\operatorname{S}(z_{1},z_{2})=\sum_{h,k\geqslant
0}\frac{1}{z_{1}^{h+1}z_{2}^{k+1}}\int_{\Omega}x^{h}\,y^{k}\,\omega(x,y)\operatorname{d}x\operatorname{d}y=\sum_{h,k\geqslant
0}\,\frac{\omega_{h,k}}{z_{1}^{h+1}z_{2}^{k+1}}.$

We introduce the _first $k$-marginal series_ for the Stieltjes function
(2.17),

$\displaystyle\operatorname{S}_{1,k}(z)=\sum_{h=0}^{+\infty}\,\frac{\omega_{h,k}}{z^{h+1}},$
$\displaystyle k\geqslant 0,$ (2.18)

and the _second $h$-marginal series_,

$\displaystyle\operatorname{S}_{2,h}(z)=\sum_{k=0}^{+\infty}\,\frac{\omega_{h,k}}{z^{k+1}},$
$\displaystyle h\geqslant 0.$ (2.19)

We observe that the functions $\operatorname{S}_{1,k}(z)$,
$\operatorname{S}_{2,h}(z)$, are Stieltjes function in one variable associated
with the moments

$\displaystyle\int_{\Omega}x^{h}\,y^{k}\,\omega(x,y)\operatorname{d}x\operatorname{d}y=\omega_{h,k},$
$\displaystyle h,k\geqslant 0.$

##  3\. The 2D Toda lattices

Now, we will show that the bivariate orthogonal polynomials modelize the $2$D
Toda lattice. We suppose that the positive definite measure defined on a
region $\Omega\subset\mathbb{R}^{2}$ is given in terms of a weight function in
the form:

$\displaystyle\operatorname{d}\mu(x,y)=\omega(x,y)\operatorname{d}x\operatorname{d}y,$

and suppose that all the moments in (2.8) exist. Let
$\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ be the monic OPS associated
to the measure $\operatorname{d}\mu(x,y)$.

From now on we consider a weight function, $\omega$, with an _evolution term_
$t\geqslant 0$, i.e. $\omega(x,y,t)$ such that
$\omega(x,y,0)\equiv\omega(x,y)$. Observe that the explicit shape of the
weight function depending on the evolution term $t$ is unknown.

We also suppose that the time dependent weight function $\omega(x,y,t)$ has
finite moments, i.e.

$\displaystyle\omega_{h,k}(t)=\int_{\Omega}x^{h}\,y^{k}\,\omega(x,y,t)\operatorname{d}x\operatorname{d}y<+\infty,$
$\displaystyle h,k\geqslant 0,$ $\displaystyle t\geqslant 0,$

exist, and is normalized such that $\omega_{0,0}(t)=1$.

Observe that the moments depend on the time variable $t$ and verify
$\omega_{h,k}(0)=\omega_{h,k}$. For $t\geqslant 0$, we define the inner
product,

$\displaystyle\langle
p,q\rangle_{t}=\int_{\Omega}p(x,y)\,q(x,y)\,\omega(x,y,t)\operatorname{d}x\operatorname{d}y,$
$\displaystyle p,q\in\Pi.$

Obviously,
$\langle\boldsymbol{\cdot},\boldsymbol{\cdot}\rangle_{0}=\langle\boldsymbol{\cdot},\boldsymbol{\cdot}\rangle$.

Let $\big{\\{}\mathbb{P}_{n}(t)\big{\\}}_{n\geqslant 0}$ $\equiv$
$\big{\\{}\mathbb{P}_{n}(x,y,t)\big{\\}}_{n\geqslant 0}$ be the monic
orthogonal polynomial system associated to $\omega(x,y,t)$. Clearly, these are
two variables polynomials (on $x$ and $y$) whose coefficients depend on $t$,

$\displaystyle\mathbb{P}_{n}(t)=\mathbb{X}_{n}+\operatorname{G}^{n}_{n-1}(t)\,\mathbb{X}_{n-1}+\cdots+\operatorname{G}^{n}_{0}(t)\,\mathbb{X}_{0},$
$\displaystyle n\geqslant 0,$

with $\mathbb{P}_{n}(0)=\mathbb{P}_{n}$.

As usual, we use the dot-notation for the derivative with respect to the time
variable $t$. Since $\mathbb{P}_{n}(t)$ is a monic polynomial we get

$\displaystyle\accentset{\mbox{\large.}}{\mathbb{P}}_{n}(t)=\frac{\operatorname{d}}{\operatorname{d}t}\mathbb{P}_{n}(t)=\accentset{\mbox{\large.}}{\operatorname{G}}^{n}_{n-1}(t)\,\mathbb{X}_{n-1}+\accentset{\mbox{\large.}}{\operatorname{G}}^{n}_{n-2}(t)\,\mathbb{X}_{n-2}+\cdots+\accentset{\mbox{\large.}}{\operatorname{G}}^{n}_{0}(t)\,\mathbb{X}_{0},$

i.e., $\accentset{\mbox{\large.}}{\mathbb{P}}_{n}(t)$ is a $(n+1)$ vector of
polynomials of degree less than or equal to $n-1$.

In addition, the symmetric positive-definite matrix

$\displaystyle\operatorname{H}_{n}(t)=\langle\mathbb{P}_{n}(t),\mathbb{P}^{\top}_{n}(t)\rangle_{t},$
$\displaystyle n\geqslant 0,$

also depends on $t$, and $\operatorname{H}_{n}(0)=\operatorname{H}_{n}$.

The sequence $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ satisfy the three term
relations (2.10), but now, the matrix coefficients depend on $t$. In this way,
for $n\geqslant 0$, there exist matrices
$\operatorname{D}_{n,i}(t),\operatorname{C}_{n,i}(t)$, of respective sizes
$(n+1)\times(n+1)$ and $(n+1)\times n$, $i=1,2$, such that

$\displaystyle\begin{aligned}
x\,\mathbb{P}_{n}(t)=\operatorname{L}_{n,1}\,\mathbb{P}_{n+1}(t)+\operatorname{D}_{n,1}(t)\,\mathbb{P}_{n}(t)+\operatorname{C}_{n,1}(t)\,\mathbb{P}_{n-1}(t),\\\
y\,\mathbb{P}_{n}(t)=\operatorname{L}_{n,2}\,\mathbb{P}_{n+1}(t)+\operatorname{D}_{n,2}(t)\,\mathbb{P}_{n}(t)+\operatorname{C}_{n,2}(t)\,\mathbb{P}_{n-1}(t),\end{aligned}$
(3.1)

where $\mathbb{P}_{-1}(t)=0$ and $\operatorname{C}_{-1}(t)=0$. Moreover,

$\displaystyle\operatorname{D}_{n,1}(t)\,\operatorname{H}_{n}(t)$
$\displaystyle=\langle
x\,\mathbb{P}_{n}(t),\mathbb{P}^{\top}_{n}(t)\rangle_{t},$
$\displaystyle\operatorname{D}_{n,2}(t)\,\operatorname{H}_{n}(t)$
$\displaystyle=\langle
y\,\mathbb{P}_{n}(t),\mathbb{P}^{\top}_{n}(t)\rangle_{t},$
$\displaystyle\operatorname{C}_{n,1}(t)\,\operatorname{H}_{n-1}(t)$
$\displaystyle=\operatorname{H}_{n}(t)\,\operatorname{L}^{\top}_{n-1,1},$
$\displaystyle\operatorname{C}_{n,2}(t)\,\operatorname{H}_{n-1}(t)$
$\displaystyle=\operatorname{H}_{n}(t)\,\operatorname{L}^{\top}_{n-1,2},$

and $\operatorname{D}_{n,i}(0)=\operatorname{D}_{n,i}$,
$\operatorname{C}_{n,i}(0)=\operatorname{C}_{n,i}$, $i=1,2$. We must remark
that the matrices $\operatorname{L}_{n,i}$ defined in (2.7) are independent of
$t$.

We also define

$\displaystyle\operatorname{D}_{n}(t)$
$\displaystyle=\operatorname{D}_{n,1}(t)+\operatorname{D}_{n,2}(t),$
$\displaystyle\operatorname{C}_{n}(t)=\operatorname{C}_{n,1}(t)+\operatorname{C}_{n,2}(t).$

In this case, the tridiagonal Jacobi matrices also depend on the $t$ variable,
and adopt the form

$\displaystyle\boldsymbol{\operatorname{J}}_{i}(t)=\begin{bmatrix}\operatorname{D}_{0,i}(t)&\operatorname{L}_{0,i}&&\\\
\operatorname{C}_{1,i}(t)&\operatorname{D}_{1,i}(t)&\operatorname{L}_{1,i}&\\\
&\operatorname{C}_{2,i}(t)&\operatorname{D}_{2,i}(t)&\ddots\\\
&&\ddots&\ddots\end{bmatrix},$ (3.2)

where the main diagonal matrices, $\operatorname{D}_{n,i}(t)$, are of square
type and with increasing size with $n$, the matrices in the subdiagonal,
$\operatorname{C}_{n,i}$, are $(n+1)\times n$ matrices, and in the
superdiagonal, the matrices $\operatorname{L}_{n,i}$ are $n\times(n+1)$
matrices defined in (2.7), and the other elements are zero matrices of
appropriate size.

We define the infinite block matrix

$\displaystyle\boldsymbol{\operatorname{A}}(t)=\begin{bmatrix}\mathtt{0}&&&\\\
\operatorname{C}_{1}(t)&\mathtt{0}&&\\\
&\operatorname{C}_{2}(t)&\mathtt{0}&\\\ &&\ddots&\ddots\end{bmatrix},$ (3.3)

where $\mathtt{0}$ are square zero matrices of adequate size.

In the next theorem, we prove several characterizations for 2D Toda lattices,
relating the orthogonality and the shape of the weight function with the
evolution term. We will omit the $t$ variable when it can be understood by the
context.

######  Theorem 3.1.

Let $w(x,y,t)$ be a normalized weight function defined on a domain
$\Omega\subset\mathbb{R}^{2}$, and let
$\big{\\{}\mathbb{P}_{n}(t)\big{\\}}_{n\geqslant 0}$ be the corresponding
monic orthogonal polynomial system. The following statements are equivalent:

  1. (i)

The sequences $\big{\\{}\operatorname{D}_{n,i}(t)\big{\\}}_{n\geqslant 0}$ and
$\big{\\{}\operatorname{C}_{n,i}(t)\big{\\}}_{n\geqslant 1}$, for $i=1,2$,
satisfy the $2$D Toda lattice

$\displaystyle\begin{cases}\accentset{\mbox{\large.}}{\operatorname{D}}_{n,i}(t)=\operatorname{C}_{n}(t)\,\operatorname{L}_{n-1,i}-\operatorname{L}_{n,i}\,\operatorname{C}_{n+1}(t),\\\
\accentset{\mbox{\large.}}{\operatorname{C}}_{n,i}(t)=\operatorname{C}_{n}(t)\,\operatorname{D}_{n-1,i}(t)-\operatorname{D}_{n,i}(t)\,\operatorname{C}_{n}(t),\end{cases}$
$\displaystyle n\geqslant 1.$ (3.4)

  2. (ii)

The infinite matrices $\boldsymbol{\operatorname{J}}_{i}(t)$ and
$\boldsymbol{\operatorname{A}}(t)$ verify the Lax-Nakamura-type pairs

$\displaystyle\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{J}}}_{i}=\big{[}\boldsymbol{\operatorname{A}},\boldsymbol{\operatorname{J}}_{i}\big{]}=\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{i}-\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{A}},$
(3.5)

for $i=1,2$, where
$\boldsymbol{\operatorname{J}}_{i}\equiv\boldsymbol{\operatorname{J}}_{i}(t)$
and
$\boldsymbol{\operatorname{A}}_{i}\equiv\boldsymbol{\operatorname{A}}_{i}(t)$
are defined in (3.2) and (3.3), respectively.

  3. (iii)

For $h,k\geqslant 0$, we have

$\displaystyle\accentset{\mbox{\large.}}{\aoverbrace[L1R]{\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}}}=\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}-\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\boldsymbol{\operatorname{A}}.$

  4. (iv)

For $h,k\geqslant 0$ the moments satisfy

$\displaystyle\accentset{\mbox{\large.}}{\omega}_{h,k}=-\omega_{h+1,k}-\omega_{h,k+1}+\omega_{h,k}\big{(}\omega_{1,0}+\omega_{0,1}\big{)}.$
(3.6)

  5. (v)

The Stieltjes function satisfies

$\displaystyle\accentset{\mbox{\large.}}{\operatorname{S}}(z_{1},z_{2},t)=\big{(}\operatorname{D}_{0}-z_{1}-z_{2}\big{)}\operatorname{S}(z_{1},z_{2},t)+\operatorname{S}_{1,0}(z_{1},t)+\operatorname{S}_{2,0}(z_{2},t).$
(3.7)

  6. (vi)

The weight function is solution of the differential equation

$\displaystyle\accentset{\mbox{\large.}}{\omega}=-\big{(}x_{1}+x_{2}-D_{0}\big{)}\,\omega.$
(3.8)

  7. (vii)

The weight function is given by

$\displaystyle\omega(x,y,t)=\dfrac{\mathrm{e}^{-(x_{1}+x_{2})t}\omega(x,y)}{\int_{\Omega}\mathrm{e}^{-(x_{1}+x_{2})t}\omega(x,y)\operatorname{d}x\operatorname{d}y}.$
(3.9)

  8. (viii)

The MOPS satisfy

$\accentset{\mbox{\large.}}{\mathbb{P}}_{n}(t)=\operatorname{C}_{n}(t)\,\mathbb{P}_{n-1}(t),$
(3.10)

for $n\geqslant 1$ and $t\geqslant 0$.

###### Proof.

Rewriting (3.4) in matrix notation we arrive to (3.5); and then, we get that,
(i) $\Rightarrow$ (ii). Now, we will prove by induction that (ii)
$\Rightarrow$ (iii). For $n=1$, (ii) is the induction hypothesis. Applying an
inductive reasoning we get,

$\displaystyle\accentset{\mbox{\large.}}{\aoverbrace[L1R]{\boldsymbol{\operatorname{J}}_{i}^{n}}}$
$\displaystyle=\accentset{\mbox{\large.}}{\aoverbrace[L1R]{\boldsymbol{\operatorname{J}}_{i}^{n-1}}}\boldsymbol{\operatorname{J}}_{i}+\boldsymbol{\operatorname{J}}_{i}^{n-1}\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{J}}}_{i}$
$\displaystyle=(\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{i}^{n-1}-\boldsymbol{\operatorname{J}}_{i}^{n-1}\,\boldsymbol{\operatorname{A}})\boldsymbol{\operatorname{J}}_{i}+\boldsymbol{\operatorname{J}}_{i}^{n-1}(\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{i}-\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{A}})$
by the hypotesis
$\displaystyle=\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{i}^{n}-\boldsymbol{\operatorname{J}}_{i}^{n}\,\boldsymbol{\operatorname{A}},$
$\displaystyle i=1,2.$

Now, we can see that

$\displaystyle\accentset{\mbox{\large.}}{\aoverbrace[L1R]{\boldsymbol{\operatorname{J}}_{1}^{h}\boldsymbol{\operatorname{J}}_{2}^{k}}}$
$\displaystyle=\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{J}}}_{1}^{h}\boldsymbol{\operatorname{J}}_{2}^{k}+\boldsymbol{\operatorname{J}}_{1}^{h}\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{J}}}_{2}^{k}$
$\displaystyle=\big{(}\boldsymbol{\operatorname{A}}\boldsymbol{\operatorname{J}}_{1}^{h}-\boldsymbol{\operatorname{J}}_{1}^{h}\boldsymbol{\operatorname{A}}\big{)}\boldsymbol{\operatorname{J}}_{2}^{k}+\boldsymbol{\operatorname{J}}_{1}^{h}\big{(}\boldsymbol{\operatorname{A}}\boldsymbol{\operatorname{J}}_{2}^{k}-\boldsymbol{\operatorname{J}}_{2}^{k}\boldsymbol{\operatorname{A}}\big{)}$
by the last identity
$\displaystyle=\boldsymbol{\operatorname{A}}\big{(}\boldsymbol{\operatorname{J}}_{1}^{h}\boldsymbol{\operatorname{J}}_{2}^{k}\big{)}-\big{(}\boldsymbol{\operatorname{J}}_{1}^{h}\boldsymbol{\operatorname{J}}_{2}^{k}\big{)}\boldsymbol{\operatorname{A}},$
$\displaystyle h,k\geqslant 0.$

To prove that (iii) $\Rightarrow$ (iv), we recall that the moments

$\displaystyle\omega_{h,k}(t)=\int_{\Omega}x^{h}\,y^{k}\,\omega(x,y,t)\operatorname{d}x\operatorname{d}y$

depend on $t$, and by (2.16), are written in the form

$\displaystyle\omega_{h,k}(t)=\ell_{0}^{\top}\big{(}\boldsymbol{\operatorname{J}}_{1}(t)\big{)}^{h}\,\big{(}\boldsymbol{\operatorname{J}}_{2}(t)\big{)}^{k}\ell_{0}.$

Taking derivatives and using (iii), we get

$\displaystyle\accentset{\mbox{\large.}}{\omega}_{h,k}$
$\displaystyle=\ell_{0}^{\top}\,\accentset{\mbox{\large.}}{\aoverbrace[L1R]{\boldsymbol{\operatorname{J}}_{1}^{h}\,\
\boldsymbol{\operatorname{J}}_{2}^{k}}}\,\ell_{0}=\ell_{0}^{\top}\,\big{(}\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}-\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\boldsymbol{\operatorname{A}}\big{)}\,\ell_{0}=-\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{1}\,\operatorname{C}_{1}$
$\displaystyle=-\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{0}\,\operatorname{D}_{0}-\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{1}\,\operatorname{C}_{1}+\ell_{0}^{\top}\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}\,\ell_{0}\,\operatorname{D}_{0}$
$\displaystyle=-\omega_{h+1,k}-\omega_{h,k+1}+\omega_{h,k}\operatorname{D}_{0}$
$\displaystyle=-\omega_{h+1,k}-\omega_{h,k+1}+\omega_{h,k}\,\big{(}\omega_{1,0}+\omega_{0,1}\big{)},$

using the explicit expression of the matrix $\boldsymbol{\operatorname{A}}$
given in (3.3) and the special shape of the matrix
$\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}$.
Here, $\ell_{1}=\begin{bmatrix}0&1&1&0&\cdots\end{bmatrix}$, and together with
$\ell_{0}^{\top}$ allows access to the $(0,1)$ block matrix element in the
matrix
$\boldsymbol{\operatorname{J}}_{1}^{h}\,\boldsymbol{\operatorname{J}}_{2}^{k}$.

From the definition of the Stieltjes function, and taking into account their
time dependence, we can write using (iv) that

$\displaystyle\operatorname{S}(z_{1},z_{2},t)=\sum_{h,k=0}^{+\infty}\,\frac{\omega_{h,k}(t)}{z_{1}^{h+1}z_{2}^{k+1}}.$

Now, computing the _dot_ -derivative, and applying (3.6), we successively get
that

$\displaystyle\accentset{\mbox{\large.}}{\operatorname{S}}(z_{1},z_{2},t)=\sum_{h,k=0}^{+\infty}\,\frac{\accentset{\mbox{\large.}}{\omega}_{h,k}}{z_{1}^{h+1}z_{2}^{k+1}}$
$\displaystyle\hskip
14.22636pt=-\sum_{h,k=0}^{+\infty}\,\frac{\omega_{h+1,k}}{z_{1}^{h+1}z_{2}^{k+1}}-\sum_{h,k=0}^{+\infty}\,\frac{\omega_{h,k+1}}{z_{1}^{h+1}z_{2}^{k+1}}+\operatorname{D}_{0}\,\sum_{h,k=0}^{+\infty}\,\frac{\omega_{h,k}}{z_{1}^{h+1}z_{2}^{k+1}}$
$\displaystyle\hskip
14.22636pt=-\sum_{h=1,k=0}^{+\infty}\,\frac{\omega_{h,k}}{z_{1}^{h}z_{2}^{k+1}}-\sum_{h=0,k=1}^{+\infty}\,\frac{\omega_{h,k}}{z_{1}^{h+1}z_{2}^{k}}+\operatorname{D}_{0}\,\operatorname{S}(z_{1},z_{2},t)$
$\displaystyle\hskip
14.22636pt=-z_{1}\Big{(}\operatorname{S}(z_{1},z_{2},t)-\sum_{k=0}^{+\infty}\,\frac{\omega_{0,k}}{z_{1}z_{2}^{k+1}}\Big{)}-z_{2}\Big{(}\operatorname{S}(z_{1},z_{2},t)-\sum_{h=0}^{+\infty}\,\frac{\omega_{h,0}}{z_{1}^{h+1}z_{2}}\Big{)}$
$\displaystyle\hskip
270.30118pt+\operatorname{D}_{0}\,\operatorname{S}(z_{1},z_{2},t)$
$\displaystyle\hskip
14.22636pt=-\big{(}z_{1}+z_{2}-\operatorname{D}_{0}\big{)}\,\operatorname{S}(z_{1},z_{2},t)+\sum_{h=0}^{+\infty}\,\frac{\omega_{h,0}}{z_{1}^{h+1}}+\sum_{k=0}^{+\infty}\,\frac{\omega_{0,k}}{z_{2}^{k+1}}$
$\displaystyle\hskip
14.22636pt=-\big{(}z_{1}+z_{2}-\operatorname{D}_{0}\big{)}\,\operatorname{S}(z_{1},z_{2},t)+\operatorname{S}_{1,0}(z_{1},t)+\operatorname{S}_{2,0}(z_{2},t),$

by using the marginal Stieltjes functions defined in (2.18)-(2.19). Therefore,
the bivariate Stieltjes function $\operatorname{S}(x,y,t)$ satisfies the _dot_
-differential equation (3.7), and so we arrive to (v).
Departing from (v), taking into account the representation (2.17) for the
Stieltjes function associated with $\omega$, and equation (3.7), we have

$\int_{\Omega}\frac{\accentset{\mbox{\large.}}{\omega}(x,y,t)}{(z_{1}-x)(z_{2}-y)}\,\operatorname{d}x\operatorname{d}y=-\int_{\Omega}\frac{(z_{1}-x)\omega(x,y,t)}{(z_{1}-x)(z_{2}-y)}\,\operatorname{d}x\operatorname{d}y\\\
-\int_{\Omega}\frac{(z_{2}-y)\omega(x,y,t)}{(z_{1}-x)(z_{2}-y)}\,\operatorname{d}x\operatorname{d}y-\int_{\Omega}\frac{(x+y-\operatorname{D}_{0})\omega(x,y,t)}{(z_{1}-x)(z_{2}-y)}\,\operatorname{d}x\operatorname{d}y\\\
+\operatorname{S}_{1}(z_{1},t)+\operatorname{S}_{2}(z_{2},t).$

Now, from (2.18) and (2.19) we see that the functions $\operatorname{S}_{1}$
and $\operatorname{S}_{2}$ have the following integral representation

$\displaystyle\operatorname{S}_{1}(z_{1},t)$
$\displaystyle=\int_{\Omega}\frac{\omega(x,y,t)}{z_{1}-x}\,\operatorname{d}x\operatorname{d}y,$
$\displaystyle\operatorname{S}_{2}(z_{1},t)=\int_{\Omega}\frac{\omega(x,y,t)}{z_{2}-y}\,\operatorname{d}x\operatorname{d}y.$

Hence the above equation takes the form

$\displaystyle\int_{\Omega}\frac{\accentset{\mbox{\large.}}{\omega}(x,y,t)+(x+y-\operatorname{D}_{0})}{(z_{1}-x)(z_{2}-y)}\,\operatorname{d}x\operatorname{d}y=0,$

and so we get (3.8) (by the analiticity of this function).
Now, departing from (vi) and solving (3.8), we get

$\displaystyle\omega(x,y,t)=\kappa^{-1}\mathrm{e}^{-(x_{1}+x_{2}-\omega_{10}-\omega_{01})t}\omega(x,y)$
with $\displaystyle\omega_{00}=1,$

with $\kappa$ given by
$\displaystyle\kappa=\int_{\Omega}\mathrm{e}^{-(x_{1}+x_{2}-\omega_{10}-\omega_{01})t}\omega(x,y)\operatorname{d}x\operatorname{d}y$.
Using this expression for $\kappa$ we get the representation (3.9) for
$\omega$.
(vii) $\Rightarrow$ (viii): As $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant
0}$ is a basis of the linear space of two variable polynomials we know that,
there exist matrices $\alpha_{0},\alpha_{1},\ldots,\alpha_{n-1}$ (that could
depend on $t$, but not on $x,y$) such that

$\displaystyle\accentset{\mbox{\large.}}{\mathbb{P}}_{n}=\alpha_{0}\,\mathbb{P}_{n-1}+\alpha_{1}\,\mathbb{P}_{n-2}+\cdots+\alpha_{n-1}\,\mathbb{P}_{0}.$

Using the orthogonality conditions we get that

$\displaystyle\alpha_{k}H_{n-1-k}=\int_{\Omega}\accentset{\mbox{\large.}}{\mathbb{P}}_{n}\,\mathbb{P}_{n-1-k}^{\top}\,\omega\,\operatorname{d}x\operatorname{d}y,$
$\displaystyle k\in\big{\\{}0,1,\ldots,n-1\big{\\}}.$

We determine the Fourier coefficients in the above expansion as

$\displaystyle\int_{\Omega}\accentset{\mbox{\large.}}{\mathbb{P}}_{n}\,\mathbb{P}_{n-1-k}^{\top}\omega\operatorname{d}x\operatorname{d}y$
$\displaystyle=\int_{\Omega}{\mathbb{P}}_{n}\,\accentset{\mbox{\large.}}{\mathbb{P}}_{n-1-k}^{\top}\omega\operatorname{d}x\operatorname{d}y+\int_{\Omega}{\mathbb{P}}_{n}\,\mathbb{P}_{n-1-k}^{\top}\,\accentset{\mbox{\large.}}{\omega}\operatorname{d}x\operatorname{d}y$
$\displaystyle=-\int_{\Omega}{\mathbb{P}}_{n}\,\mathbb{P}_{n-1-k}^{\top}\,(x+y-D_{0})\,\omega\,\operatorname{d}x\operatorname{d}y$
$\displaystyle=-\int_{\Omega}{\mathbb{P}}_{n}\,\mathbb{P}_{n-1-k}^{\top}\,(x+y)\omega\,\operatorname{d}x\operatorname{d}y.$

Now, using the orthogonality we get that $\alpha_{k}\equiv\mathtt{0}$, the
null matrix, for $k=1,\ldots,n-1$, and from (2.11), i.e.

$\displaystyle(x+y)\,\mathbb{P}_{n}=\operatorname{L}_{n}\,\mathbb{P}_{n+1}+\operatorname{D}_{n}\,\mathbb{P}_{n}+\operatorname{C}_{n}\,\mathbb{P}_{n-1},$

we get that

$\displaystyle\alpha_{0}H_{n-1}=\operatorname{C}_{n}H_{n-1},$ i.e.
$\displaystyle\alpha_{0}=\operatorname{C}_{n}.$

From here the result follows.
We close the diagram by differentiating the first relation in (3.1)

$\displaystyle
x\,\accentset{\mbox{\large.}}{\mathbb{P}}_{n}=\operatorname{L}_{n,1}\,\accentset{\mbox{\large.}}{\mathbb{P}}_{n+1}+\accentset{\mbox{\large.}}{\operatorname{D}}_{n,1}\,\mathbb{P}_{n}+\operatorname{D}_{n,1}\,\accentset{\mbox{\large.}}{\mathbb{P}}_{n}+\accentset{\mbox{\large.}}{\operatorname{C}}_{n,1}\,\mathbb{P}_{n-1}+\operatorname{C}_{n,1}\,\accentset{\mbox{\large.}}{\mathbb{P}}_{n-1},$

and using (3.10), i.e., we apply the hypotesis (viii),

$x\,\operatorname{C}_{n}\mathbb{P}_{n-1}=\operatorname{L}_{n,1}\,\operatorname{C}_{n+1}\mathbb{P}_{n}+\accentset{\mbox{\large.}}{\operatorname{D}}_{n,1}\,\mathbb{P}_{n}+\operatorname{D}_{n,1}\,\operatorname{C}_{n}\mathbb{P}_{n-1}\\\
+\accentset{\mbox{\large.}}{\operatorname{C}}_{n,1}\,\mathbb{P}_{n-1}+\operatorname{C}_{n,1}\,\operatorname{C}_{n-1}\mathbb{P}_{n-2},$

and again, we use (3.1) for $n-1$, to deduce

$\operatorname{C}_{n}(\operatorname{L}_{n-1,1}\,\mathbb{P}_{n}+\operatorname{D}_{n-1,1}\,\mathbb{P}_{n-1}+\operatorname{C}_{n-1,1}\,\mathbb{P}_{n-2})\\\
=(\operatorname{L}_{n,1}\,\operatorname{C}_{n+1}+\accentset{\mbox{\large.}}{\operatorname{D}}_{n,1})\mathbb{P}_{n}+(\operatorname{D}_{n,1}\,\operatorname{C}_{n}+\accentset{\mbox{\large.}}{\operatorname{C}}_{n,1})\mathbb{P}_{n-1}+\operatorname{C}_{n,1}\,\operatorname{C}_{n-1}\mathbb{P}_{n-2}.$

Since $\big{\\{}\mathbb{P}_{n}\big{\\}}_{n\geqslant 0}$ is an orthogonal
basis, matching the coefficients, (3.4) follows for $i=1$.

The case $i=2$ is analogous. ∎

As a corollary of this theorem we get a representation for the Stieltjes
function associated with $\omega$ that modelizes the Toda lattice, as

$\operatorname{S}(z_{1},z_{2},t)=e^{-(z_{1}+z_{2})t+\widetilde{d}(t)}\Big{(}K-\int_{0}^{t}\big{(}\operatorname{S}_{1,0}(z_{1},s)\\\
+\operatorname{S}_{2,0}(z_{2},s)\big{)}e^{(z_{1}+z_{2})s-\widetilde{d}(s)}\operatorname{d}s\Big{)},$

where $K$ is a positive constant and
$\widetilde{d}(t)=\int_{0}^{t}\operatorname{D}_{0}(s)\operatorname{d}s$.

Following [8], we suppose that the matrix coefficients of the three term
relation satisfy the $2$D Toda lattice

$\displaystyle\begin{cases}\accentset{\mbox{\large.}}{\operatorname{D}}_{n,i}(t)=\operatorname{C}_{n}(t)\,\operatorname{L}_{n-1,i}-\operatorname{L}_{n,i}\,\operatorname{C}_{n+1}(t),\\\
\accentset{\mbox{\large.}}{\operatorname{C}}_{n,i}(t)=\operatorname{C}_{n,i}(t)\,\operatorname{D}_{n-1}(t)-\operatorname{D}_{n}(t)\,\operatorname{C}_{n,i}(t),\end{cases}$
(3.11)

for $i=1,2$, and $n\geqslant 0$, and then, summing both equations, we get the
system

$\displaystyle\begin{cases}\accentset{\mbox{\large.}}{\operatorname{D}}_{n}(t)=\operatorname{C}_{n}(t)\,\operatorname{L}_{n-1}-\operatorname{L}_{n}\,\operatorname{C}_{n+1}(t),\\\
\accentset{\mbox{\large.}}{\operatorname{C}}_{n}(t)=\operatorname{C}_{n}(t)\,\operatorname{D}_{n-1}(t)-\operatorname{D}_{n}(t)\,\operatorname{C}_{n}(t).\end{cases}$

Notice that, in equations (3.11) appear $\operatorname{C}_{n}(t)$ and
$\operatorname{C}_{n,i}(t)$. We prove that the Toda system deduced in [8] is
equivalent to our Toda system (3.4).

######  Proposition 3.2.

The $2$D Toda lattice (3.11) is equivalent to (3.4).

###### Proof.

We transform (3.11) by using (2.14),

$\displaystyle\accentset{\mbox{\large.}}{\operatorname{C}}_{n,1}(t)$
$\displaystyle=\operatorname{C}_{n,1}(t)\,\operatorname{D}_{n-1}(t)-\operatorname{D}_{n}(t)\,\operatorname{C}_{n,1}(t)$
$\displaystyle=\operatorname{C}_{n,1}(t)(\operatorname{D}_{n-1,1}(t)+\operatorname{D}_{n-1,2}(t))-(\operatorname{D}_{n,1}(t)+\operatorname{D}_{n,2}(t))\operatorname{C}_{n,1}(t)$
$\displaystyle=(\operatorname{C}_{n,1}(t)+\operatorname{C}_{n,2}(t))\operatorname{D}_{n-1,1}(t)-\operatorname{C}_{n,2}(t)\operatorname{D}_{n-1,1}(t)$
$\displaystyle\hskip
49.79231pt+\operatorname{C}_{n,1}(t)\operatorname{D}_{n-1,2}(t)-\operatorname{D}_{n,1}(t)(\operatorname{C}_{n,1}(t)+\operatorname{C}_{n,2}(t))$
$\displaystyle\hskip
128.0374pt+\operatorname{D}_{n,1}(t)\operatorname{C}_{n,2}(t)-\operatorname{D}_{n,2}(t)\operatorname{C}_{n,1}(t)$
$\displaystyle=\operatorname{C}_{n}(t)\,\operatorname{D}_{n-1,1}(t)-\operatorname{D}_{n,1}(t)\,\operatorname{C}_{n}(t).$

In this way, (3.4) is deduced. ∎

##  4\. Isospectrality of $\boldsymbol{\operatorname{J}}_{i}(t)$

In this Section, we study the eigenvalues of the matrix
$\boldsymbol{\operatorname{J}}_{i}(t)$.

######  Proposition 4.1.

The eigenvalues of the matrix $\boldsymbol{\operatorname{J}}_{i}(t)$ are real,
for $i=1,2.$

###### Proof.

By (2.15), we get that

$\displaystyle\boldsymbol{\operatorname{H}}^{1/2}\,\boldsymbol{\operatorname{L}}_{i}\,\boldsymbol{\operatorname{H}}^{-1/2}=\boldsymbol{\operatorname{J}}_{i},$
$\displaystyle i=1,2,$

i.e., $\boldsymbol{\operatorname{L}}_{i}$ and
$\boldsymbol{\operatorname{J}}_{i}$ are similar and have the same eigenvalues.
Since $\boldsymbol{\operatorname{L}}_{i}$ is self-adjoint and real, then all
of its eigenvalues are real, and then, the eigenvalues of
$\boldsymbol{\operatorname{J}}_{i}$ are. ∎

Next, we end the paper by proving that the matrices
$\boldsymbol{\operatorname{J}}_{i}(t)$ are isospectral, i.e., their
eigenvalues are independent of the time variable $t$, as occurs in the
univariate case for standard Lax pairs (cf. [16]).

######  Theorem 4.2.

Let $\boldsymbol{\operatorname{J}}_{i}(t)$ be given by (2.13). Then,
$\boldsymbol{\operatorname{J}}_{i}(t)$, $i=1,2$, satisfies (3.5) if, and only
if, the algebraic spectrum of $\boldsymbol{\operatorname{J}}_{i}(t)$, $i=1,2$,
is independent of $t$.

###### Proof.

From (2.10) written in matrix notation as

$\displaystyle
x_{i}\,\boldsymbol{\operatorname{P}}=\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{P}},$
$\displaystyle i=1,2,$

taking derivatives, and using (3.5) we get successively

$\displaystyle\accentset{\mbox{\large.}}{x}_{i}\,\boldsymbol{\operatorname{P}}+x_{i}\,\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{P}}}$
$\displaystyle=\big{(}\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{J}}_{i}-\boldsymbol{\operatorname{J}}_{i}\,\boldsymbol{\operatorname{A}}\big{)}\,\boldsymbol{\operatorname{P}}+\boldsymbol{\operatorname{J}}_{i}\,\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{P}}}$
$\displaystyle\accentset{\mbox{\large.}}{x}_{i}\,\boldsymbol{\operatorname{P}}$
$\displaystyle=\big{(}\boldsymbol{\operatorname{J}}_{i}-x_{i}\,\operatorname{I}\big{)}\,\big{(}-\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{P}}+\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{P}}}\big{)}.$

Now, from Theorem 3.1 we know that the Lax–Nakamura equation for
$\boldsymbol{\operatorname{J}}_{i}$, i.e. (3.5), is equivalent to (3.10), i.e.

$\displaystyle\accentset{\mbox{\large.}}{\mathbb{P}}_{n}(t)=\operatorname{C}_{n}(t)\,\mathbb{P}_{n-1}(t),$
$\displaystyle n\in\mathbb{N},$

which in matrix notation reads as

$\displaystyle\accentset{\mbox{\large.}}{\boldsymbol{\operatorname{P}}}=\boldsymbol{\operatorname{A}}\,\boldsymbol{\operatorname{P}},$
hence, we get that,
$\displaystyle\accentset{\mbox{\large.}}{x}_{i}\,\boldsymbol{\operatorname{P}}=\operatorname{0},$

and so, $\displaystyle\accentset{\mbox{\large.}}{x}_{i}=0$. ∎

## References

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# Combinatorial proof of Maslov index formula in Heegaard Floer Theory

Roman Krutowski  Department of Mathematics, University of California, Los
Angeles<EMAIL_ADDRESS>
###### Abstract.

We prove Lipshitz’s Maslov index formula in Heegaard Floer homology via the
combinatorics of Heegaard diagrams.

##  1\. Introduction

In [7, 8] Peter Ozsváth and Zoltán Szabó introduced Heegaard Floer homology, a
collection of invariants for closed oriented 3-manifolds. Since then Heegaard
Floer homology has emerged as an extremely powerful invariant, producing
strong results in low-dimensional topology (see [7, 8, 10, 9, 4, 1, 5]). All
of the numerous versions of Heegaard Floer homology involve counting the
number of points in the unparametrized moduli space of $J$-holomorphic disks
of a certain Maslov index joining some intersection points of two half-
dimensional tori in a certain symmetric product of a surface.

The most notable advantage of Heegaard Floer homology when compared to other
types of Floer Homology is its combinatorial nature which allows computation
of these invariants in various cases. One of the ingredients is the
combinatorial formula for Maslov index $\mu(\varphi)$ of a Whitney disk
$\varphi$, which is the Fredholm index of some differential operator, or
alternatively, a homology class of a certain loop in the Lagrangian
Grassmanian. In [11], Jacob Rasmussen gave a formula that relates the
intersection number of the disk with the fat diagonal in the symmetric
product, with its Maslov index. Later on, Robert Lipshitz in the paper devoted
to the cylindrical reformulation of the whole theory [2] gave a purely
combinatorial formula for the Maslov index of these disks.

The proof of this formula in [2, 3] is based on an elegant geometric approach.
In this paper, we provide a combinatorial proof of this formula which is
inspired by the proof of the index formula for Maslov $n$-gons due to Sucharit
Sarkar [12].

Let $(\Sigma,\bm{\alpha},\bm{\beta})$ be a Heegaard diagram. Any Whitney disk
$\varphi\in\pi_{2}(\bm{x},\bm{y})$ connecting $\bm{x},\bm{y}\in
Sym^{g}(\Sigma)$ has a _shadow_ $D(\varphi)$ (see Definition 2.1), a certain
$2$-chain in $\Sigma$ with boundary satisfying $\partial(\partial
D(\varphi)\cap\bm{\alpha})=\bm{y}-\bm{x}$ and $\partial(\partial
D(\varphi)\cap\bm{\beta})=\bm{x}-\bm{y}$. We denote the set of such $2$-chains
by $\mathcal{D}(\bm{x},\bm{y})$ and call them _domains_. We denote by
$\mathcal{D}$ the set of all domains in all Heegaard diagrams (see Section 2
for details).

In this paper, we assume that $\bm{\alpha}$ curves intersect $\bm{\beta}$
curves perpendicularly with respect to some metric on $\Sigma$. Among the
domains, there are two special types that serve us as building blocks.
_Bigons_ and _rectangles_ are $2$-sided and $4$-sided domains, respectively,
which are homeomorphic to disks with all angles equal to $90^{\circ}$ and
which do not contain any $\bm{x}$\- and $\bm{y}$-coordinates in the interior.

Given two domains $D\in\mathcal{D}(\bm{x},\bm{y})$ and
$D^{\prime}\in\mathcal{D}(\bm{y},\bm{z})$ the composition of these domains
$D*D^{\prime}$ is a domain $D+D^{\prime}\in\mathcal{D}(\bm{x},\bm{z})$.

In this paper any map $\overline{\mu}\colon\mathcal{D}\rightarrow\mathbb{Z}$
will be called an _index_. An index $\overline{\mu}$ is said to be _additive_
if for any Heegaard diagram and any two of its domains
$D\in\mathcal{D}(\bm{x},\bm{y})$ and $D^{\prime}\in\mathcal{D}(\bm{y},\bm{z})$
the following holds

$\overline{\mu}(D*D^{\prime})=\overline{\mu}(D)+\overline{\mu}(D^{\prime}).$

In Section 2 we introduce two types of transformations of any Heegaard diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$ which assign to each of its domain $D$ a new
domain $D^{\prime}$ in the new Heegaard diagram. These transformations are
called _finger moves_ and _empty stabilizations_. An index $\overline{\mu}$ is
said to be _stable_ if for any such transformation
$\overline{\mu}(D)=\overline{\mu}(D^{\prime})$.

Define the _combinatorial index_ $\widetilde{\mu}$ of a domain
$D\in\mathcal{D}(\bm{x},\bm{y})$ via the formula due to Lipshitz [2]

(1)
$\widetilde{\mu}(D)\coloneqq\widetilde{\mu}_{\bm{x}}(D)+\widetilde{\mu}_{\bm{y}}(D)+e(D).$

where $\widetilde{\mu}_{\bm{x}}(D)$ is a point measure of $D$ at $\bm{x}$ and
$e(D)$ is the Euler measure of $D$ (see Subsection 2.1).

We are now ready to formulate our main results.

######  Theorem 1.1.

There exists a unique index
$\overline{\mu}\colon\mathcal{D}\rightarrow\mathbb{Z}$ satisfying the
following axioms:

  1. (1)

$\overline{\mu}$ is additive;

  2. (2)

$\overline{\mu}$ is stable;

  3. (3)

$\overline{\mu}(B)=1$ for any bigon $B\in\mathcal{D}$;

  4. (4)

$\overline{\mu}(R)=1$ for any rectangle $R\in\mathcal{D}$.

Moreover, this index agree with the combinatorial index $\widetilde{\mu}$.

######  Theorem 1.2.

Let $(\Sigma,\bm{\alpha},\bm{\beta})$ be a Heegaard diagram and $\varphi$ be a
Whitney disk connecting two generators of the corresponding Heegaard Floer
chain complex. Then

$\mu(\varphi)=\widetilde{\mu}(D(\varphi)),$

where $\mu(\varphi)$ is the Maslov index of $\varphi$.

Acknowledgments. I would like to thank Sucharit Sarkar for his guidance and
encouragement, Ko Honda for helpful conversations and suggestions, and Gleb
Terentiuk for valuable comments.

##  2\. Notations and preliminary results

###  2.1. Heegaard Floer Theory preliminaries

To start, recall the definition of a Heegaard diagram. Let $\Sigma$ be an
oriented Riemannian surface (i.e., there is a fixed Riemannian metric) and
$\bm{\alpha}=\\{\alpha_{1},\ldots,\alpha_{g}\\}$ and
$\bm{\beta}=\\{\beta_{1},\ldots,\beta_{g}\\}$ be two sets of non-intersecting
_oriented_ 111This is nonstandard but useful to assume for the entirety of the
paper. simple closed curves such that both $\bm{\alpha}$ and $\bm{\beta}$
generate half-dimensional subspaces of $H_{1}(\Sigma)$ (in particular this
means that $g$ is not smaller than the genus $g(\Sigma)$ of the surface). The
last part is equivalent to assuming that the complement of $\bm{\alpha}$
consists of $g-g(\Sigma)+1$ components (the same holds for $\bm{\beta})$. We
also assume that the $\bm{\alpha}$ and $\bm{\beta}$ curves intersect
perpendicularly. We call by _regions_ closures of connected components of
$\Sigma\setminus(\bm{\alpha}\cup\bm{\beta})$. The collection
$(\Sigma,\bm{\alpha},\bm{\beta})$ is usually called an _unpointed Heegaard
diagram_ , but in the text we refer to it simply as a _Heegaard diagram_. We
need not assume these diagrams to be pointed, because it is not relevant for
the Maslov index calculations.

The Heegaard Floer homology chain complex of a diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$ is generated by $g$-tuples of points of the
form $\bm{x}=\\{x_{1},\ldots,x_{g}\\}$ where
$x_{i}\in\alpha_{i}\cap\beta_{\sigma(i)}$ and $\sigma\in S_{g}$ is arbitrary.
We may regard $\bm{x}$ as a point in $Sym^{g}(\Sigma)$ which belongs to
$T_{\bm{\alpha}}\cap T_{\bm{\beta}}$ where
$T_{\bm{\alpha}}=\alpha_{1}\times\alpha_{2}\times\ldots\times\alpha_{g}$ and
$T_{\bm{\beta}}=\beta_{1}\times\beta_{2}\times\ldots\times\beta_{g}$.

Let as consider the unit disk $D^{2}$ in $\mathbb{C}$ with the usual
orientation. Let $s_{1}\subset\partial D^{2}$ be the portion of the oriented
boundary that connects $i$ to $-i$ and let $s_{2}\subset\partial D^{2}$ be the
remaining portion of the boundary connecting $-i$ to $i$. The differentials
and chain maps are signed counts of $J$-holomorphic maps
$u:D^{2}\setminus\\{i,-i\\}\rightarrow Sym^{g}(\Sigma)$ representing Whitney
disks in $\pi_{2}(\bm{x},\bm{y})$. Here a Whitney disk is a homotopy type of
maps from $D^{2}$ to $Sym^{g}(\Sigma)$ such that $s_{1}$ is mapped to
$T_{\bm{\alpha}}$, $s_{2}$ is mapped to $T_{\bm{\beta}}$ and $i$, $-i$ are
sent to $\bm{x}$ and $\bm{y}$, respectively.

Let $D$ be a $2$-chain obtained as a sum of regions of a Heegaard diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$ with integer coefficients, $D=\sum_{R}a(R)R$
where $a(R)\in\mathbb{Z}$. Such a $2$-chain $D$ is called a _domain_ if there
exist two generators $\bm{x}$ and $\bm{y}$ such that $\partial(\partial
D|_{\bm{\alpha}})\coloneqq\partial(\partial D\cap\bm{\alpha})=\bm{y}-\bm{x}$
(here we regard $\bm{x}$ and $\bm{y}$ as $0$-chains in $\Sigma$) and
$\partial(\partial D|_{\bm{\beta}})=\bm{x}-\bm{y}$. We say that $D$ _connects_
$\bm{x}$ and $\bm{y}$ and denote the set of these domains by
$\mathcal{D}(\bm{x},\bm{y})$. We also denote by $\mathcal{D}$ the set of all
pairs of $((\Sigma,\bm{\alpha},\bm{\beta}),D)$, where the first element in the
pair is some Heegaard diagram and the second is some domain in this Heegaard
diagram.

######  Definition 2.1.

Given a Whitney disk $\varphi\in\pi_{2}(\bm{x},\bm{y})$ we associate a domain
$D(\varphi)\in\mathcal{D}(\bm{x},\bm{y})$ called _shadow_ of $\varphi$ as
follows: to a region $R$ we assign the number $n_{R}(\varphi)$ which is equal
to the intersection number $\varphi\cdot Z_{r}$, where $Z_{r}=\\{r\\}\times
Sym^{g-1}(\Sigma)$ for some $r$ in the interior of $R$ (note that
$\varphi\cdot Z_{r}$ does not depend on a choice of $r$). Then set

$D(\varphi)=\sum_{R}n_{R}(\varphi)R\in\mathcal{D}(\bm{x},\bm{y}).$

Let us recall the definition of the Maslov index of a Whitney disk
$\varphi\in\pi_{2}(\bm{x},\bm{y})$ where $\bm{x}$ and $\bm{y}$ are some
generators of the chain complex associated with a given Heegaard diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$. The generators $\bm{x}$ and $\bm{y}$ are
regarded as points belonging to $T_{\bm{\alpha}}\cap T_{\bm{\beta}}\subset
Sym^{g}(\Sigma)$. Let $Gr_{g}$ be a space of totally real $g$-dimensional
subspaces of $\mathbb{C}^{g}$. Since $\varphi$ is regarded as a map from
$D^{2}\setminus\\{i,-i\\}=[0,1]\times\mathbb{R}$ to $Sym^{g}(\Sigma)$ we take
the pullback of $TSym^{g}(\Sigma)$ to $D^{2}$ which is a trivial bundle
$\mathbb{C}^{g}$. At point $\bm{x}$ we pick a “short” path $\gamma_{\bm{x}}$
between $T_{\bm{x}}(T_{\bm{\alpha}})$ and $T_{\bm{x}}(T_{\bm{\beta}})$ in
$Gr_{g}$ and also construct in the same fashion a path $\gamma_{\bm{y}}$ for
$y$ (by “short” we mean a path in $\mathcal{P}^{-}(Gr_{g})$ in Seidel’s
terminology, see [14, Section 11]). The map from ${0}\times\mathbb{R}$ to
$T_{\bm{\alpha}}$ assigns a path $\gamma_{0}$ in $Gr_{g}$ by considering
pullback of $T(T_{\bm{\alpha}})\subset TSym^{g}(\Sigma)$. Analogously, we get
a path $\gamma_{1}$. Finally, the Maslov index $\mu(\varphi)$ is equal to a
composition of all these four paths

$\mu(\varphi)=[\gamma_{\bm{x}}\circ\gamma_{0}\circ\gamma_{\bm{y}}\circ\gamma_{1}]\in
H_{1}(Gr_{g})=\mathbb{Z}.$

We also introduce _combinatorial index_ of any domain
$D\in\mathcal{D}(\bm{x},\bm{y})$. For $x_{i}\in\bm{x}$ define
$\widetilde{\mu}_{x_{i}}(D)$ as the average of the coefficients of $D$ in the
$4$ regions to which $x_{i}$ belongs. Then the _point measure_ of $D$ at
$\bm{x}$ is $\widetilde{\mu}_{x}(D)=\sum_{i=1}^{g}\widetilde{\mu}_{x_{i}}(D)$.
For a $2$-chain $D$ the Euler measure $D$ is $\frac{1}{2\pi}$ of the integral
of the curvature of the metric of $\Sigma$ over $D$. It is equal to the
$2$-cochain that assigns $\frac{1}{2}(2-n)$ to a $2n$-gon region. For a domain
$D\in\mathcal{D}(\bm{x},\bm{y})$ we assign its combinatorial index by

$\widetilde{\mu}(D)=\widetilde{\mu}_{\bm{x}}(D)+\widetilde{\mu}_{\bm{y}}(D)+e(D).$

###  2.2. Transformations of Heegaard diagrams

We introduce two types of transformations on a given Heegaard diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$ that are used in this paper.

######  Definition 2.2.

Let $\gamma$ be an oriented path in $\Sigma$ which is transverse to
$\bm{\alpha}$ and $\bm{\beta}$ and is disjoint from
$\bm{\alpha}\cap\bm{\beta}$; see Figure 2. Also assume the endpoints $p_{0}$
and $p_{1}$ of $\gamma$ do not belong to $\bm{\alpha}\cup\bm{\beta}$. Let $U$
be a small neighborhood of $\gamma$. Let $\psi$ be an isotopy of $\Sigma$
supported on a small neighborhood of $\gamma$ which moves $\bm{\alpha}$ curves
in the direction of $\gamma$ as given in Figure 2. The _finger move_ on the
$\bm{\alpha}$ curves along the curve $\gamma$ is a restriction
$\psi|_{\alpha}$; this does not move the $\bm{\beta}$ curves. Analogously we
define the _finger move on the $\bm{\beta}$ curves_.

A transition from a Heegaard diagram $(\Sigma,\bm{\alpha},\bm{\beta})$ to the
diagram $(\Sigma,\psi(\bm{\alpha}),\bm{\beta})$ will be called by _performing
a finger move on $\bm{\alpha}$ along $\gamma$_. Finger moves were introduced
in [13].

######  Definition 2.3.

Given a domain $D$, its _image_ $D^{\prime}=\psi(D)$ under a finger move
$\psi$ is defined as follows: first decompose a finger move $\psi$ into a
sequence $\psi=\psi_{1}\circ\ldots\circ\psi_{m}$ of finger moves where under
any $\psi_{i}$ only two new points of intersection between $\bm{\alpha}$ and
$\bm{\beta}$ appear (see Figure 2). Let $R_{1}$, $R_{2}$ and $R_{3}$ be
regions at which $\psi_{1}\circ\ldots\circ\psi_{i-1}(D)$ has coefficients
$a,b$ and $c$, respectively, and let the finger move $\psi_{i}$ be as shown in
the picture. Then for new regions
$R_{1}^{\prime},R_{2}^{\prime},R_{3}^{\prime},R_{4}^{\prime}$ and
$R_{5}^{\prime}$ shown in Figure 2 the coefficients are $a,b,c,b$ and $a+c-b$
as shown. Repeating this procedure gives us $\psi(D)$.

(a)

(b)

Figure 1. Finger move

(a)

(b)

Figure 2. Image of a domain.

######  Definition 2.4.

Let $D$ be a domain of a Heegaard diagram $(\Sigma,\bm{\alpha},\bm{\beta})$.
We call an _empty stabilization_ of this Heegard diagram with respect to $D$ a
stabilization which is obtained by taking the connected sum with the standard
genus $1$ Heegaard diagram for $S^{3}$ where the attaching disk belongs to a
region of $(\Sigma,\bm{\alpha},\bm{\beta})$ where the coefficient of $D$ is
equal to $0$. The new Heegaard diagram
$(\Sigma^{\prime},\bm{\alpha}^{\prime},\bm{\beta}^{\prime})$ is also called an
_empty stabilization_ of $(\Sigma,\bm{\alpha},\bm{\beta})$. The image of $D$
under an empty stabilization is $D$ itself in the new diagram.

If we are given a Whitney disk $\varphi\in\pi_{2}(\bm{x},\bm{y})$ we may
define its _image_ $\varphi^{\prime}$ under a finger move by isotoping
$\varphi$ with the accordance to the induced isotopy of $T_{\bm{\alpha}}$
inside $Sym^{g}(\Sigma)$.

We say that the stabilization is empty with respect to a Whitney disk
$\varphi$ if it is empty with respect to $D(\varphi)$. For the empty
stabilization with respect to $\varphi$ the image of $\varphi$ is just
$\varphi\times z$ where $z$ is the intersection point of added $\alpha_{g+1}$
and $\beta_{g+1}$.

We will extensively use the invariance of the Maslov index and of $\mu$ under
these two types of transformations of a Heegaard diagram which is shown in the
lemma below.

######  Lemma 2.5.

Let $D\in\mathcal{D}(\bm{x},\bm{y})$ be a domain in Heegaard diagram
$(\Sigma,\bm{\alpha},\bm{\beta})$ and let $\varphi\in\pi_{2}(\bm{x},\bm{y})$
be a Whitney disk. Let $D^{\prime}$ be the image of $D$ under a finger move or
an empty stabilization with respect to $D$. Let $\varphi^{\prime}$ be the
image of $\varphi$ under a finger move or an empty stabilization with respect
to $\varphi$. Then $\mu(\varphi)=\mu(\varphi^{\prime})$ and
$\widetilde{\mu}(D)=\widetilde{\mu}(D^{\prime})$.

###### Proof.

First, consider the case when this transformation is an empty stabilization
with respect to $D$ (or $\varphi)$. Let $\alpha_{g+1}$ and $\beta_{g+1}$ be
two new curves of $(\Sigma,\bm{\alpha}^{\prime},\bm{\beta}^{\prime})$ and $z$
be their point of intersection. We have $\bm{x}^{\prime}=\bm{x}\cup\\{z\\}$,
$\bm{y}^{\prime}=\bm{y}\cup\\{z\\}$ and
$D^{\prime}=D\in\mathcal{D}(\bm{x}^{\prime},\bm{y}^{\prime})$ (or
$\varphi^{\prime}\in\pi_{2}(\bm{x},\bm{y}))$. First,
$\widetilde{\mu}(D)=\widetilde{\mu}(D^{\prime})$, because
$\widetilde{\mu}_{z}(D)=0$. As for the Maslov index, notice that paths
$\gamma_{0}$ and $\gamma_{1}$ in $Gr_{g}$ rise to paths $\gamma_{0}^{\prime}$
and $\gamma_{1}^{\prime}$ in $Gr_{g+1}$ obtained by taking direct sum with
one-dimensional real subspaces of $\mathbb{R}^{2(g+1)}=\mathbb{C}^{g+1}$
corresponding to $T_{z}\alpha_{g+1}$ and $T_{z}\beta_{g+1}$ respectively.
Hence $\mu(\varphi)=\mu(\varphi^{\prime})$.

If the transformation is a finger move then the Maslov indices are equal
because finger move keeps $\bm{\alpha}$ and $\bm{\beta}$ unchanged near
$\bm{x}$ and $\bm{y}$. Hence, inspecting the definition of Maslov indices we
see that only the paths in $Gr_{g}$ coming from the boundary $\partial D^{2}$
differ, and they are isotopic to the initial ones, so the class in
$H_{1}(Gr_{g})$ does not change. Finally,
$\widetilde{\mu}(D)=\widetilde{\mu}(D^{\prime})$ is immediate from the formula
(see also [12, Theorem 3.4]). ∎

###  2.3. Additivity of indices

Our strategy for proving Lipshitz’s formula is based on decomposing any domain
into the composition of trivial pieces: bigons and rectangles. To apply this
decomposition we need the additivity of the Maslov index $\mu$ and of the
combinatorial index $\widetilde{\mu}$ under the composition of domains. We
know that the Maslov index $\mu$ is additive. It was shown in more generality
in [12] that $\widetilde{\mu}$ is additive and here we repeat the proof in our
terms.

######  Definition 2.6.

Let $\gamma_{1}$ and $\gamma_{2}$ be oriented $1$-chains in $\Sigma$
intersecting transversely. Denote by $\gamma_{1}\cdot\gamma_{2}$ intersection
number of these $1$-chains, which is equal to the signed count of intersection
points where the sign is defined by comparing the orientation of $\Sigma$ and
orientation coming from $\gamma_{1}$ and $\gamma_{2}$. A contribution to
intersection number at an endpoint is given by a fraction $\pm\frac{1}{2}$ or
$\pm\frac{1}{4}$ as in [12].

######  Lemma 2.7.

Let $D\in\pi_{2}(\bm{x},\bm{y})$ and $D^{\prime}$ be any other $2$-chain in
the Heegaard diagram $(\Sigma,\bm{\alpha},\bm{\beta})$. Then
$\widetilde{\mu}_{\bm{x}}(D^{\prime})-\widetilde{\mu}_{\bm{y}}(D^{\prime})=\partial
D^{\prime}|_{\bm{\alpha}}\cdot\partial D|_{\bm{\beta}}=\partial
D^{\prime}|_{\bm{\beta}}\cdot\partial D|_{\bm{\alpha}}$.

###### Proof.

Given orientations on $\bm{\alpha}$ at each intersection point $p$ between
$\bm{\alpha}$ and $\bm{\beta}$ we give numbers to regions from $\mathit{I}$ to
$\mathit{IV}$. Namely, $\mathit{I}$ and $\mathit{II}$ quadrants are in the
upper half of $\bm{\beta}$ (the half in the positive direction of
$\bm{\alpha}$) and $\mathit{I}$st quadrant is the one for which the
orientation induced from $\Sigma$ is opposite to the orientation of
$\bm{\alpha}$ at $p$. Quadrants $\mathit{III}$ and $\mathit{IV}$ are defined
follow $\mathit{II}$ in the counterclockwise order on $\Sigma$.

Let us pick a point $p$ in the first quadrant near the point $x_{i}\in\bm{x}$
lying on some $\alpha_{k}$ and travel parallely to $\partial
D|_{\bm{\alpha}_{k}}$ until we reach a first quadrant near $y_{j}\in\bm{y}$
where $\partial(\partial D|_{\bm{\alpha}_{k}})=y_{j}-x_{i}$. Then let us track
the coefficient of $D^{\prime}$ at each point along this path. Each time we
change the region passing through $\bm{\beta}$ this coefficient changes by the
intersection number of this small portion of our path with $\partial
D^{\prime}|_{\bm{\beta}}$. Hence, the difference between the coefficients is
equal to $\partial D|_{\alpha_{k}}\cdot\partial D^{\prime}|_{\bm{\beta}}$.

Now, by taking such paths for all points in $\bm{x}$ and all $4$ quadrants for
each of these points and then averaging we get
$\widetilde{\mu}_{\bm{x}}(D^{\prime})-\widetilde{\mu}_{\bm{y}}(D^{\prime})=\partial
D|_{\bm{\alpha}}\cdot\partial D^{\prime}|_{\bm{\beta}}$ ∎

######  Corollary 2.8.

Let $D\in\pi_{2}(\bm{x},\bm{y})$ and $D^{\prime}\in\pi_{2}(\bm{y},\bm{z})$.
Then
$\widetilde{\mu}(D*D^{\prime})=\widetilde{\mu}(D)+\widetilde{\mu}(D^{\prime})$.

###### Proof.

First, we easily see that

$\widetilde{\mu}(D*D^{\prime})-\widetilde{\mu}(D)-\widetilde{\mu}(D^{\prime})=\widetilde{\mu}_{\bm{x}}(D^{\prime})+\widetilde{\mu}_{\bm{z}}(D)-\widetilde{\mu}_{\bm{y}}(D^{\prime})-\widetilde{\mu}_{\bm{y}}(D).$

Second, applying Lemma 2.7 we get
$\widetilde{\mu}_{\bm{x}}(D^{\prime})-\widetilde{\mu}_{\bm{y}}(D^{\prime})=\partial
D|_{\bm{\alpha}}\cdot\partial D^{\prime}|_{\bm{\beta}}$ and
$\widetilde{\mu}_{\bm{y}}(D)-\widetilde{\mu}_{\bm{z}}(D)=\partial
D|_{\bm{\alpha}}\cdot\partial D^{\prime}|_{\bm{\beta}}$.

Hence

$\widetilde{\mu}(D*D^{\prime})-\widetilde{\mu}(D)-\widetilde{\mu}(D^{\prime})=0.$

∎

##  3\. Main theorem

######  Theorem 3.1.

For a given domain $D$ in a Heegaard diagram $(\Sigma,\bm{\alpha},\bm{\beta})$
there is a sequence of finger moves and empty stabilizations such that in the
new Heegaard diagram the image of $D$ can be represented as a composition of
bigons, rectangles and their negatives.

Before proving this theorem we show how Theorems 1.1 and 1.2 follow from
Theorem 3.1.

###### Proof of Theorem 1.1.

Let $D\in\mathcal{D}(\bm{x},\bm{y})$. Apply Theorem 3.1 to obtain the image
$D^{\prime}$ of $D$ and the decomposition $D^{\prime}=D_{1}*\ldots*D_{k}$ of
$D^{\prime}$ in the new Heegaard diagram. Here each $D_{i}$ is either a bigon,
a rectangle, or the negative of a bigon or a rectangle. Since $\overline{\mu}$
is additive we may compute
$\overline{\mu}(D^{\prime})=\overline{\mu}(D_{1})+\ldots+\overline{\mu}(D_{k})$.
For each $D_{i}$ the value $\overline{\mu}(D_{i})$ can be easily inferred from
the $4$ axioms. ∎

###### Proof of Theorem 1.2.

We are given a Whitney disk $\varphi\in\pi_{2}(\bm{x},\bm{y})$ and we may
assume that $g(\Sigma)>1$ by applying an empty stabilization if necessary. By
Theorem 3.1 there is a transformation of the given Heegaard diagram such that
there is a decomposition $D(\varphi^{\prime})=D_{1}*\ldots*D_{k}$, where
$\varphi^{\prime}$ is an image of $\varphi$ under this transformation;
equivalently $0=D(\varphi^{\prime})*(-D_{1})*\ldots*(-D_{k})$. Since each
$D_{i}$ is either a bigon or a rectangle (possibly negative) there exists a
corresponding Whitney disk $\varphi_{i}$ such that $D(\varphi_{i})=-D_{i}$.
Then
$\varphi^{\prime}*\varphi_{1}*\ldots*\varphi_{k}\in\pi_{2}(\bm{x},\bm{x})$
and, moreover, $D(\varphi^{\prime}*\varphi_{1}*\ldots*\varphi_{k})=0$. From
[7, Proposition 2.15] it now follows that
$\varphi^{\prime}*\varphi_{1}*\ldots*\varphi_{k}=0\in\pi_{2}(\bm{x},\bm{x})$,
so $\mu(\varphi^{\prime}*\varphi_{1}*\ldots*\varphi_{k})=0$. Hence,
$\mu(\varphi^{\prime})=-(\mu(\varphi_{1})+\ldots+\mu(\varphi_{k}))$.

It now suffices to show that $\mu(B)=\widetilde{\mu}(B)=1$ and
$\mu(R)=\widetilde{\mu}(R)=1$ for any bigon $B$ and any rectangle $R$. For a
bigon $\mu(B)=1$ as it agrees with the dimension of the (regular) moduli space
of biholomorphisms from $B$ to the strip $[0,1]\times\mathbb{R}$. The fact
that $\mu(R)=1$ is shown in [7, Section 8.4]. Finally,
$\widetilde{\mu}(R)=\widetilde{\mu}(B)=1$ by direct computation. ∎

Now we prove the main theorem.

###### Proof of Theorem 3.1.

In the proof we abuse notation and denote the image of a domain $D$ under any
empty stabilization or a finger move by same letter $D$ .

We break our proof into several steps starting with a general domain in Step 1
and simplifying it gradually by transformations of Heegaard diagrams. Before
we start dealing with general domains we need additional preparation which we
perform at Step 0.

Step 0. Quadrilaterals.

(a)

(b)

Figure 3. $270^{\circ}$ angle

Before considering the most general domains we need to take care of another
type of “building block”. Namely, let $D$ be an embedded domain with non-
negative coefficients which is homeomorphic to a disk and has a boundary
consisting of four sides. We would call such $D$ a _quadrilateral_. If all
angles of $D$ are $90^{\circ}$ then it is already a rectangle. Otherwise, an
angle at one of its vertices $a$ is $270^{\circ}$. Consider a finger move
shown in Figure 3 which creates two bigons $B_{1}\in\pi_{2}(a,b)$ and
$B_{2}\in\pi_{2}(b,c)$. Then $D*(-B_{1})$ is a quadrilateral with one more
$90^{\circ}$ angle than $D$. Proceeding in the same fashion we decompose $D$
into a rectangle and several bigons.

Henceforth, in the following steps it is enough to decompose any domain into
quadrilaterals and bigons with arbitrary angles.

Step 1. Making the boundary embedded.

First, we show that after several finger moves we may compose $D$ with some
bigons (or their negatives) and obtain a domain $D^{\prime}$ with embedded
boundary.

(a)

(b)

Figure 4. Making the boundary embedded

Assume without loss of generality that $\partial(\partial
D|_{\alpha_{1}})=y_{1}-x_{1}$ with respect to the chosen orientation on
$\alpha_{1}$. Here $x_{1}$ lies on $\beta_{1}$ and $y_{1}$ lies on $\beta_{2}$
(where $\beta_{2}$ may be equal to $\beta_{1}$). We may assume that the
positively oriented embedded arc from $x_{1}$ to $y_{1}$ in $\alpha_{1}$ is
covered $k+1$ times by $\partial D$ and the positively oriented embedded arc
from $y_{1}$ to $x_{1}$ is covered $k$ times. For now assume $k>0$.

Since the $\bm{\alpha}$ and $\bm{\beta}$ curves are oriented, for any region
we may distinguish whether it lies to the left of a given $\bm{\alpha}$ or
$\bm{\beta}$ curve. Let $a$ be a point very close to $x_{1}$ on $\beta_{1}$ to
the left of $\alpha_{1}$ (see Figure 4). Also let $b$ be a point on
$\alpha_{1}$ near $y_{1}$ not belonging to the positive arc from $x_{1}$ to
$y_{1}$. Let $\gamma$ be (a slight extension of) a curve starting at $a$ and
parallel to $\alpha_{1}$ until it hits $\beta_{2}$ and then we connect it with
$b$. We make a finger move on $\bm{\beta}$ curves along the curve $\gamma$
creating new points of intersection of $\bm{\beta}$ curves intersecting
$\gamma$ (including $\beta_{1}$ and $\beta_{2}$ ) with $\alpha_{1}$. Let $z$
be the new intersection point of $\beta_{1}$ and $\alpha_{1}$ which is the
closest to $x_{1}$ on $\beta_{1}$.

As a result, we created a bigon $B$ connecting $x_{1}$ to $z$. Then
$D^{\prime}=D*(-B)\in\mathcal{D}(\\{z,x_{2},\dots,x_{g}\\},\\{y_{1},y_{2},\ldots,y_{g}\\})$.
Note that in $\partial D^{\prime}$ the arc from $z$ to $y_{1}$ is covered $k$
times and the arc from $y_{1}$ to $z$ is covered $(k-1)$ times.

In the case $k<0$ we would draw $\gamma$ parallel to the negative embedded arc
from $x_{1}$ to $y_{1}$ and proceed analogously. Repeating this procedure we
first make sure that $\partial D\cap\bm{\alpha}$ is embedded and then we
repeat it for $\bm{\beta}$ and end up with a domain having an embedded
boundary.

Step 2. Making the boundary connected.

(a)

(b)

Figure 5. Making boundary connected

Assume the boundary of a domain $D$ consists of more than one closed curve as
shown in Figure 5. Suppose $\partial D\cap(\alpha_{1}\cap\beta_{1})=y_{1}$,
$\partial D\cap(\beta_{1}\cap\alpha_{2})=x_{2}$ and $\partial(\partial
D|_{\alpha_{2}})=y_{2}-x_{2}$. Let us pick a point on $\alpha_{1}$ close to
$y_{1}$ and let $\gamma$ be an arc connecting it to a point on the restriction
of some other component of $\partial D$ to $\bm{\beta}$ such that
$int(\gamma)$ does not intersect $\partial D$. We denote by
$\beta_{1}^{\prime}$ the curve to which the endpoint of $\gamma$ belongs and
$\partial(\partial D|_{\beta_{1}^{\prime}})=x_{2}^{\prime}-y_{1}^{\prime}$
while $\partial(\partial
D|_{\alpha_{1}^{\prime}})=y_{1}^{\prime}-x_{1}^{\prime}$. We also can make
sure that $\gamma$ intersects $\beta_{1}^{\prime}$ at a point near
$y_{1}^{\prime}$. We may decompose $\gamma=\gamma_{1}\gamma_{2}\gamma_{3}$
where $\gamma_{1}$ and $\gamma_{2}$ are small portions near the ends and
$\gamma_{2}$ is the remaining portion in the middle.

We draw a curve $\gamma_{1}^{\prime}$ connecting a point near $y_{1}$ on
$\beta_{1}$ with some point near the end of $\gamma_{1}$ such that
$\gamma_{1}^{\prime}$ does not intersect $\gamma_{1}$. Starting at the end of
$\gamma_{1}^{\prime}$, we draw a curve $\gamma_{2}^{\prime}$ parallel to
$\gamma_{2}$. Then we connect the end of $\gamma_{2}$ by a curve $\gamma_{3}$
parallel to $\beta_{1}^{\prime}$ with a point on $\alpha_{1}^{\prime}$ near
$y_{1}^{\prime}$. We set
$\gamma^{\prime}=\gamma_{1}^{\prime}\gamma_{2}^{\prime}\gamma_{3}^{\prime}$.

Now we make a finger move on $\bm{\alpha}$ curves along $\gamma$ and we also
make a finger move on $\bm{\beta}$ curves along $\gamma^{\prime}$. Denote the
new points of intersection between $\alpha_{1}$ and $\beta_{1}^{\prime}$ by
$a$ and $b$, and analogously denote the points of intersection between
$\beta_{1}$ and $\alpha_{1}^{\prime}$ by $c$ and $d$.

Denote the new quadrilateral by
$D^{\prime}\in\mathcal{D}(\\{b,c\\},\\{y_{1},y_{1}^{\prime}\\})$ and the two
new bigons by $B_{1}\in\mathcal{D}(\\{a\\},\\{b\\})$ and
$B_{2}\in\mathcal{D}(\\{c\\},\\{d\\})$.

Then consider

$D*(-D^{\prime})*B_{1}*B_{2}\in\mathcal{D}(\bm{x},\\{a,y_{2}^{\prime},...,d,y_{2},...\\})$

whose boundary contains one fewer component than $D$. From here we may proceed
inductively on the number of boundary components.

(a)

(b)

Figure 6. Decreasing the number of sides

Step 3. Reducing to a quadrangle or a bigon boundary. Let $D$ be a domain and
applying Steps 1 and 2 we assume that $\partial D$ is embedded and connected.

Let the boundary of a domain $D$ be $2n$-sided, i.e. is of the form
$x_{1}y_{1}x_{2}y_{2}x_{3}\ldots y_{n}$ with $n>2$ where $x_{i}y_{i}$ is an
arc on $\bm{\alpha}$ and $y_{i}x_{i+1}$ is an arc on $\bm{\beta}$. Assume that
the first $4$ sides lie on $\alpha_{1}$, $\beta_{1}$, $\alpha_{2}$ and
$\beta_{2}$ (see Figure 6). Let us pick a point $y_{1}^{\prime}$ on
$x_{1}y_{1}$ near $y_{1}$ and a point $y_{2}^{\prime}$ on $\beta_{2}$ near
$y_{2}$. Let $\gamma$ be an arc connecting $y_{1}^{\prime}$ and
$y_{2}^{\prime}$ such that $int(\gamma)\cap\partial D=\emptyset$ and the arcs
$\gamma$, $y_{1}y_{1}^{\prime}$, $y_{1}x_{2}$, $x_{2}y_{2}$ and
$y_{2}y_{2}^{\prime}$ bound an embedded disk. We make a finger move on
$\bm{\alpha}$ curves along $\gamma$ and denote by $a$ and $b$ two new points
of intersection between $\alpha_{1}$ and $\beta_{2}$. Notice that we created
two new domains: a bigon $B\in\mathcal{D}(\\{a\\},\\{b\\})$ and a
quadrilateral $D^{\prime}\in\mathcal{D}(\\{x_{2},b\\},\\{y_{1},y_{2}\\})$.

We may replace $D$ with a domain

$D*(-D^{\prime})*B\in\pi_{2}(\\{x_{1},x_{2},\ldots,x_{n}\\},\\{a,x_{2},y_{3},\ldots,y_{n}\\}),$

which has $2n-2$ sides.

(a)

(b)

Figure 7. 4-sided boundary.

Step 4a. 4-sided boundary.

Given two domains which have the same 4-sided curve as the boundary, their
difference is an element in $H_{2}(\Sigma)$ which is generated by $[\Sigma]$.
We will prove in Claim 3.2 that $\Sigma$ can be decomposed into bigons and
rectangles after some finger moves and empty stabilizations. Therefore we may
add $k[\Sigma]$ to a given domain $D$ to ensure that we get a domain
represented by an embedded $2$-chain. This domain, which we also call $D$, is
isotopic to a disk with $m$ handles.

We will now show how to decompose this handlebody $D$ into bigons and
quadrilaterals.

(a)

(b)

Figure 8. 2-sided boundary

We depict handles as pairs of circles with opposite orientations and one of
the handles is drawn in Figure 7. We now draw two “palm trees” as in the
picture by making two finger moves on $\bm{\beta}$ curves and we assume that
the other $m-1$ handles are in the white region “between” two palm trees. Let
the $8$ new points of intersection between the $\bm{\alpha}$ and $\bm{\beta}$
curves be labeled as in Figure 7.

We now have $4$ new bigons $B_{11},B_{12},B_{21}$ and $B_{22}$ and $2$ new
quadrilaterals $R_{x}$ and $R_{y}$. Then the domain
$D*(-R_{x})*(-R_{y})*B_{11}*B_{12}*B_{21}*B_{22}\in\mathcal{D}(\\{z_{1}^{\prime},z_{2}^{\prime}\\},\\{w_{1}^{\prime},w_{2}^{\prime}\\})$
has $m-1$ handles and from here proceed inductively on the number of handles.

Step 4b. 2-sided boundary. Here we reduce this case to the $4$-sided boundary
case treated in Step 4a.

As in the $4$-sided boundary case, we may assume $D$ to be an embedded 2-chain
represented by a handlebody as shown in Figure 8. By the method of Step 0 we
may assume that both angles are $90^{\circ}$.

We may assume that $g\geqslant 2$ by applying an empty stabilization with
respect to $D$ if necessary. Let
$D\in\mathcal{D}(\\{x,z_{1},\ldots,z_{g-1}\\},\\{y,z_{1},\ldots,z_{g-1}\\})$.
We may suppose $z_{1}\in\alpha_{2}\cap\beta_{2}$ lies outside of $D$ by
applying an empty stabilization with respect to a given $D$. Let $\gamma$ be
an arc connecting the point $z_{1}$ with some point $a$ on the boundary of $D$
such that $\gamma$ intersects $D$ once at the point $a$. We may assume $a$ to
be somewhere on $\beta_{1}$. We also alter the path $\gamma$ slightly so that
it starts at some point on $\alpha_{2}$ and $\beta_{2}$ is to its left near
this endpoint.

Now draw $\gamma^{\prime}$ almost parallel to $\gamma$ so that one of its
endpoints is on $\beta_{2}$ near $z_{1}$. When $\gamma^{\prime}$ reaches
neighborhood of $a$ we extend it parallel to $-\beta_{2}$ until it hits
$\alpha_{1}$ at some point outside of $D$ near $y$.

We make a finger move on $\bm{\alpha}$ along $\gamma$ and a finger move on
$\bm{\beta}$ along $\gamma^{\prime}$ creating points
$b\in\alpha_{2}\cap\beta_{1}$ and $c\in\beta_{2}\cap\alpha_{1}$. There is now
a new quadrilateral
$R\in\mathcal{D}(\\{y,z_{1},z_{2},\ldots,z_{g-1}\\},\\{b,c,\ldots,z_{g-1}\\})$.

Therefore, we may replace $D$ with
$D*R\in\mathcal{D}(\\{x,z_{1},z_{2},\ldots,z_{g-1}\\},\\{b,c,\ldots,z_{g-1}\\})$
reducing to Step 4a.

Step 5. Inner points.

(a)

(b)

Figure 9. Inner points

By now we have represented the initial $D$ as a composition of domains which
geometrically look like bigons and rectangles, but some of the rectangles may
contain some points from generators inside of them. Namely, let
$R\in\pi_{2}({\bm{x}},{\bm{y}})$ where $\bm{x}=\\{x_{1},x_{2},a_{1},\ldots
a_{k},b_{k+1},\ldots,b_{g-2}\\}$ and $\bm{y}=\\{y_{1},y_{2},a_{1},\ldots
a_{k},b_{k+1},\ldots,b_{g-2}\\}$ where the $a_{i}$’s lie inside of $R$ and the
$b_{j}$’s lie outside of $R$.

Let $a_{1}\in\alpha_{3}\cap\beta_{3}$. We make finger moves on $\alpha_{3}$
and $\beta_{3}$ creating $2$ points of intersection between $\alpha_{3}$ and
$\beta_{1},\beta_{2}$ and $2$ points of intersection between $\beta_{3}$ and
$\alpha_{1},\alpha_{2}$ (actually, we created twice more points, but we stress
our attention on these $4$). We call these points $p_{1},p_{2},q_{1},q_{2}$,
respectively. This is illustrated in Figure 9.

Then we may see that $R=R^{\prime}*R^{\prime\prime}*R^{\prime\prime\prime}$
where

$R^{\prime}\in\mathcal{D}(\bm{x},\\{q_{1},x_{2},p_{2},a_{2}\ldots
a_{k},b_{k+1},\ldots,b_{g}\\}),$
$R^{\prime\prime}\in\mathcal{D}(\\{q_{1},x_{2},p_{2},a_{2}\ldots
a_{k},b_{k+1},\ldots,b_{g}\\},\\{q_{1},y_{2},p_{1},\ldots
a_{k},b_{k+1},\ldots,b_{g}\\}),$
$R^{\prime\prime\prime}\in\mathcal{D}(\\{q_{1},x_{2},p_{1},a_{2}\ldots
a_{k},b_{k+1},\ldots,b_{g}\\},\bm{y}).$

Each of these $3$ rectangles has fewer points from generators inside than $R$
and we may proceed by induction on the number of points.

This completes the proof of Theorem 3 assuming Claim 3.2 below. ∎

######  Claim 3.2.

The surface $\Sigma\in\mathcal{D}(\bm{x},\bm{x})$ for
${\bm{x}}=\\{x_{1},\ldots,x_{g}\\}$ is a domain which can be decomposed into
bigons and rectangles after applying a sequence of finger moves.

###### Proof.

Let us assume $g>1$. We may apply a diffeomorphism $\varphi$ to the Heegaard
diagram $(\Sigma,\bm{\alpha},\bm{\beta})$ such that the image of $\bm{x}$ is a
collection $\bm{x}^{\prime}$ such that all $x_{i}^{\prime}$ are located in
some small disk region such that $x_{1}^{\prime}$ and $x_{2}^{\prime}$ are on
the boundary of the convex hull of $\bm{x}^{\prime}$. Let us denote by
$(\Sigma,\bm{\alpha}^{\prime},\bm{\beta}^{\prime})$ the image of the initial
diagram under $\varphi$, i.e. $\bm{\alpha}^{\prime}=\varphi(\bm{\alpha})$ and
$\bm{\beta}^{\prime}=\varphi(\bm{\beta})$. We may assume that
$x_{1}^{\prime}\in\alpha_{1}^{\prime}\cap\beta_{2}^{\prime}$ and
$x_{2}^{\prime}\in\alpha_{2}^{\prime}\cap\beta_{1}^{\prime}$. Then we make
finger moves on $\alpha_{1}^{\prime},\alpha_{2}^{\prime}$ and on
$\beta_{1}^{\prime},\beta_{2}^{\prime}$ as in Figure 10 creating a rectangle
$R$ with boundary $x_{1}^{\prime}y_{1}^{\prime}x_{2}^{\prime}y_{2}^{\prime}$
such that all other points of $\bm{x}^{\prime}$ are inside of $R$. Then
$\Sigma*(-R)$ connects $\bm{x}^{\prime}$ to
$\\{y_{1}^{\prime},y_{2}^{\prime},\ldots,x_{g}^{\prime}\\}$ and it is a a
region with quadrangle boundary and $g$ handles. Now we may proceed as in Step
4a of the proof of the theorem.

Let $g=1$. We can make a finger move on $\alpha_{1}$ which creates a bigon $B$
connecting given point $x$ to some point $x^{\prime}$. Then
$(-B)*\Sigma\in\mathcal{D}(\\{x^{\prime}\\},\\{x\\})$. Now we make a
stabilization inside $B$ and can proceed as in the proof of Step 4b. ∎

(a)

(b)

Figure 10. Claim 3.2

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