# A discrete uniformization theorem for decorated piecewise Euclidean metrics
on surfaces

Xu Xu, Chao Zheng  School of Mathematics and Statistics, Wuhan University,
Wuhan, 430072, P.R.China<EMAIL_ADDRESS>School of Mathematics and
Statistics, Wuhan University, Wuhan 430072, P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce a new discretization of the Gaussian curvature on
surfaces, which is defined as the quotient of the angle defect and the area of
some dual cell of a weighted triangulation at the conic singularity. A
discrete uniformization theorem for this discrete Gaussian curvature is
established on surfaces with non-positive Euler number. The main tools are
Bobenko-Lutz’s discrete conformal theory for decorated piecewise Euclidean
metrics on surfaces and variational principles with constraints.

###### Key words and phrases:

Discrete uniformization theorem; Decorated piecewise Euclidean metrics;
Discrete Gaussian curvature; Variational principle

MSC (2020): 52C26

##  1\. Introduction

Bobenko-Lutz [2] recently introduced the decorated piecewise Euclidean metrics
on surfaces. Suppose $S$ is a connected closed surface and $V$ is a finite
non-empty subset of $S$, we call $(S,V)$ a marked surface. A piecewise
Euclidean metric (PE metric) $dist_{S}$ on the marked surface $(S,V)$ is a
flat cone metric with the conic singularities contained in $V$. A decoration
on a PE surface $(S,V,dist_{S})$ is a choice of circle of radius $r_{i}\geq 0$
at each point $i\in V$. These circles in the decoration are called vertex-
circles. We denote a decorated PE surface by $(S,V,dist_{S},r)$ and call the
pair $(dist_{S},r)$ a decorated PE metric. In this paper, we focus on the case
that $r_{i}>0$ for all $i\in V$ and each pair of vertex-circles is separated.

For a PE surface $(S,V,dist_{S})$, denote $\theta_{i}$ as the cone angle at
$i\in V$. The angle defect

(1) $W:V\rightarrow(-\infty,2\pi),\quad W_{i}=2\pi-\theta_{i},$

is used to describe the conic singularities of the PE metric. Let
$\mathcal{T}={(V,E,F)}$ be a triangulation of $(S,V)$, where $V,E,F$ are the
sets of vertices, edges and faces respectively. The triangulation
$\mathcal{T}$ for a PE surface $(S,V,dist_{S})$ is a geodesic triangulation if
the edges are geodesics in the PE metric $dist_{S}$. We use one index to
denote a vertex (such as $i$), two indices to denote an edge (such as
$\\{ij\\}$) and three indices to denote a face (such as $\\{ijk\\}$) in the
triangulation $\mathcal{T}$. For any decorated geodesic triangle $\\{ijk\\}\in
F$, there is a unique circle $C_{ijk}$ simultaneously orthogonal to the three
vertex-circles at the vertices $i,j,k$ [10]. We call this circle $C_{ijk}$ as
the face-circle of the decorated geodesic triangle $\\{ijk\\}$ and denote its
center by $c_{ijk}$ and radius by $r_{ijk}$. The center $c_{ijk}$ of the face-
circle $C_{ijk}$ of the decorated geodesic triangle $\\{ijk\\}$ is the
geometric center introduced by Glickenstein [9] and Glickenstein-Thomas [11]
for general discrete conformal structures on surfaces. Denote
$\alpha_{ij}^{k}$ as the interior intersection angle of the face-circle
$C_{ijk}$ and the edge $\\{ij\\}$. Please refer to Figure 1 (left) for the
angle $\alpha_{ij}^{k}$. The edge $\\{ij\\}$, shared by two adjacent decorated
triangles $\\{ijk\\}$ and $\\{ijl\\}$, is called weighted Delaunay if

(2) $\alpha_{ij}^{k}+\alpha_{ij}^{l}\leq\pi.$

The triangulation $\mathcal{T}$ is called weighted Delaunay in the decorated
PE metric $(dist_{S},r)$ if every edge in the triangulation is weighted
Delaunay. Connecting the center $c_{ijk}$ with the vertices $i,j,k$ by
geodesics produces a cellular decomposition of the decorated triangle
$\\{ijk\\}$. Denote $A_{i}^{jk}$ as the sum of the signed area of the two
triangles adjacent to $i$ in the cellular decomposition of the decorated
triangle $\\{ijk\\}$. The area of the triangle with the vertices $i$, $j$,
$c_{ijk}$ is positive if it is on the same side of the edge $\\{ij\\}$ as the
decorated triangle $\\{ijk\\}$, otherwise it is negative (or zero if $c_{ijk}$
lies in $\\{ij\\}$). Please refer to the shaded domain in Figure 1 (left) for
$A_{i}^{jk}$. Gluing these cells of all decorated triangles isometrically
along edges in pairs leads a cellular decomposition of the decorated PE
surface $(S,V,dist_{S},r)$. Set

$A_{i}=\sum_{\\{ijk\\}\in F}A_{i}^{jk}.$

Please refer to Figure 1 (right) for $A_{i}$.

\begin{overpic}[scale={0.4}]{figure_0.pdf} \put(15.0,15.0){$i$}
\put(80.5,15.0){$j$} \put(48.5,74.0){$k$} \put(10.0,26.0){$r_{i}$}
\put(46.0,35.0){$c_{ijk}$} \put(30.0,55.0){$\alpha_{ik}^{j}$}
\put(20.0,30.0){$\alpha_{ik}^{j}$} \put(27.0,12.5){$\alpha_{ij}^{k}$}
\put(63.0,12.5){$\alpha_{ij}^{k}$} \end{overpic}

\begin{overpic}[scale={0.6}]{figure_3.pdf} \put(48.0,42.0){$i$}
\put(92.0,44.0){$j$} \put(74.0,90.0){$k$} \put(74.0,57.0){$c_{ijk}$}
\end{overpic}

Figure 1. Domain of the signed area $A_{i}^{jk}$ in a decorated triangle
$\\{ijk\\}$ (left) and domain of the area $A_{i}$ in a decorated PE surface
(right)

######  Definition 1.1.

Suppose $(S,V,dist_{S},r)$ is a decorated PE surface and $\mathcal{T}$ is a
weighted Delaunay triangulation of $(S,V,dist_{S},r)$. The discrete Gaussian
curvature $K_{i}$ at the vertex $i\in V$ is the quotient of the angle defect
$W_{i}$ and the area $A_{i}$ of the dual cell at the vertex $i\in V$, i.e.,

(3) $K_{i}=\frac{W_{i}}{A_{i}}.$

######  Remark 1.2.

In the literature, the discrete curvature is usually defined by the angle
defect $W$ in (1). However, the angle defect $W$ is scaling invariant and does
not approximate the smooth Gaussian curvature pointwisely on smooth surfaces
as the triangulations of the surface become finer and finer. This is supported
by the discussions in [3, 7]. For the discrete Gaussian curvature $K$ in (3),
it scales by a factor $\frac{1}{u^{2}}$ upon a global rescaling of the
decorated PE metric by a factor $u$. This property is paralleling to that of
the smooth Gaussian curvature on surfaces. On the other hand, the definition
of the discrete Gaussian curvature $K_{i}$ coincides with the original
definition of the Gaussian curvature on smooth surfaces. This implies that the
discrete Gaussian curvature $K_{i}$ is a good candidate as a discretization of
the smooth Gaussian curvature on surfaces.

######  Remark 1.3.

According to Definition 1.1, the discrete Gaussian curvature $K_{i}$ defined
by (3) seems to depend on the choice of weighted Delaunay triangulations of
the decorated PE surface $(S,V,dist_{S},r)$. We will show that $K_{i}$ is an
intrinsic geometric invariant of the decorated PE surface $(S,V,dist_{S},r)$
in the sense that it is independent of the weighted Delaunay triangulations of
$(S,V,dist_{S},r)$. Note that the angle defect $W_{i}$ defined by (1) is an
intrinsic geometric invariant of a decorated PE surface, we just need to prove
that $A_{i}$ is independent of the choice of weighted Delaunay triangulations.
This is true by Lemma 2.8.

######  Remark 1.4.

The weighted Delaunay triangulation is a natural generalization of the
classical Delaunay triangulation. When the weighted Delaunay triangulation is
reduced to the classical Delaunay triangulation, i.e. $r_{i}=0$ for all $i\in
V$, the area $A_{i}$ is exactly twice the area of the Voronoi cell at the
vertex $i$. Thus the area $A_{i}$ is a generalization of the area of the
Voronoi cell at the vertex $i$. As a result, the discrete Gaussian curvature
in Definition 1.1 generalizes Kouřimská’s definition of discrete Gaussian
curvature in [17, 16].

The discrete Yamabe problem for a decorated PE metric $(dist_{S},r)$ on
$(S,V)$ asks if there exists a discrete conformal equivalent decorated PE
metric on $(S,V)$ with constant discrete Gaussian curvature. The following
discrete uniformization theorem solves this problem affirmatively for the
discrete Gaussian curvature $K$ in Definition 1.1.

######  Theorem 1.5.

For any decorated PE metric $(dist_{S},r)$ on a marked surface $(S,V)$ with
Euler number $\chi(S)\leq 0$, there is a discrete conformal equivalent
decorated PE metric with constant discrete Gaussian curvature $K$.

By the relationships of the discrete Gaussian curvature $K$ and the classical
discrete Gaussian curvature $W$, the case $\chi(S)=0$ in Theorem 1.5 is
covered by Bobenko-Lutz’s work [2]. Therefore, we just need to prove the case
$\chi(S)<0$ in Theorem 1.5.

######  Remark 1.6.

The discrete Yamabe problem on surfaces for different types of discrete
conformal structures with respect to the classical discrete Gaussian curvature
$W$ has been extensively studied in the literature. For Thurston’s circle
packings on surfaces, the solution of discrete Yamabe problem gives rise to
the famous Koebe-Andreev-Thurston Theorem. See also the work of Beardon-
Stephenson [1] for the discrete uniformization theorems for circle packings on
surfaces. For the vertex scalings introduced by Luo [18] on surfaces, Gu-Luo-
Sun-Wu [13], Gu-Guo-Luo-Sun-Wu [12], Springborn [23] and Izmestiev-Prosanov-Wu
[15] give nice answers to this problem in different background geometries.
Recently, Bobenko-Lutz [2] established the discrete conformal theory for
decorated PE metrics and prove the corresponding discrete uniformization
theorem. Since Bobenko-Lutz’s discrete conformal theory of decorated PE
metrics also applies to the Euclidean vertex scalings and thus generalizes Gu-
Luo-Sun-Wu’s result [13] and Springborn’s result [23], Theorem 1.5 also
generalizes Kouřimská’s results in [17, 16]. It should be mentioned that
Kouřimská [17, 16] constructed counterexamples to the uniqueness of PE metrics
with constant discrete Gaussian curvatures. We conjecture that the decorated
PE metric with constant discrete Gaussian curvature $K$ in Theorem 1.5 is not
unique.

The main tools for the proof of Theorem 1.5 are Bobenko-Lutz’s discrete
conformal theory for decorated PE metrics on surfaces [2] and variational
principles with constraints. The main ideas of the paper come from reading of
Bobenko-Lutz [2] and Kouřimská [17, 16].

The paper is organized as follows. In Section 2, we briefly recall Bobenko-
Lutz’s discrete conformal theory for decorated PE metrics on surfaces. Then we
show that $A_{i}$ is independent of the choice of weighted Delaunay
triangulations, i.e., Lemma 2.8. We also give some notations and a variational
characterization of the area $A_{i}^{jk}$. In this section, we also extend the
energy function $\mathcal{E}$ and the area function $A_{tot}$. In Section 3,
we translate Theorem 1.5 into an optimization problem with constraints, i.e.,
Lemma 3.2. Using the classical result from calculus, i.e., Theorem 3.3, we
translate Lemma 3.2 into Theorem 3.4. By analysing the limit behaviour of
sequences of discrete conformal factors, we get an asymptotic expression of
the function $\mathcal{E}$, i.e., Lemma 3.12. In the end, we prove Theorem
3.4.

Acknowledgements
The first author thanks Professor Feng Luo for his invitation to the workshop
“Discrete and Computational Geometry, Shape Analysis, and Applications” taking
place at Rutgers University, New Brunswick from May 19th to May 21st, 2023.
The first author also thanks Carl O. R. Lutz for helpful communications during
the workshop.

##  2\. Preliminaries on decorated PE surfaces

###  2.1. Discrete conformal equivalence and Bobenko-Lutz’s discrete conformal
theory

In this subsection, we briefly recall Bobenko-Lutz’s discrete conformal theory
for decorated PE metrics on surfaces. Please refer to Bobenko-Lutz’s original
work [2] for more details on this. The PE metric $dist_{S}$ on a PE surface
with a geodesic triangulation defines a length map
$l:E\rightarrow\mathbb{R}_{>0}$ such that $l_{ij},l_{ik},l_{jk}$ satisfy the
triangle inequalities for any triangle $\\{ijk\\}\in F$. Conversely, given a
function $l:E\rightarrow\mathbb{R}_{>0}$ satisfying the triangle inequalities
for any face $\\{ijk\\}\in F$, one can construct a PE metric on a triangulated
surface by isometrically gluing Euclidean triangles along edges in pairs.
Therefore, we use $l:E\rightarrow\mathbb{R}_{>0}$ to denote a PE metric and
use $(l,r)$ to denote a decorated PE metric on a triangulated surface
$(S,V,\mathcal{T})$.

######  Definition 2.1 ([2], Proposition 2.2).

Let $\mathcal{T}$ be a triangulation of a marked surface $(S,V)$. Two
decorated PE metrics $(l,r)$ and $(\widetilde{l},\widetilde{r})$ on
$(S,V,\mathcal{T})$ are discrete conformal equivalent if and only if there
exists a discrete conformal factor $u\in\mathbb{R}^{V}$ such that

(4) $\widetilde{r}_{i}=e^{u_{i}}r_{i},$ (5)
$\widetilde{l}_{ij}^{2}=(e^{2u_{i}}-e^{u_{i}+u_{j}})r^{2}_{i}+(e^{2u_{j}}-e^{u_{i}+u_{j}})r^{2}_{j}+e^{u_{i}+u_{j}}l_{ij}^{2}$

for all $\\{ij\\}\in E$.

######  Remark 2.2.

Note that the inversive distance

(6) $I_{ij}=\frac{l^{2}_{ij}-r^{2}_{i}-r^{2}_{j}}{2r_{i}r_{j}}$

between two vertex-circles is invariant under Möbius transformations [6].
Combining (4) and (5) gives $I=\widetilde{I}$. Since each pair of vertex-
circles is required to be separated, we have $I>1$. Therefore, Definition 2.1
can be regarded as a special case of the inversive distance circle packings
introduced by Bowers-Stephenson [4]. One can refer to [5, 14, 19, 24, 25] for
more properties of the inversive distance circle packings on triangulated
surfaces.

In general, the existence of decorated PE metrics with constant discrete
Gaussian curvatures on triangulated surfaces can not be guaranteed if the
triangulation is fixed. In the following, we work with a generalization of the
discrete conformal equivalence in Definition 2.1, introduced by Bobenko-Lutz
[2], which allows the triangulation of the marked surface to be changed under
the weighted Delaunay condition.

######  Definition 2.3 ([2], Definition 4.11).

Two decorated PE metrics $(dist_{S},r)$ and
$(\widetilde{dist}_{S},\widetilde{r})$ on the marked surface $(S,V)$ are
discrete conformal equivalent if there is a sequence of triangulated decorated
PE surfaces $(\mathcal{T}^{0},l^{0},r^{0}),...,(\mathcal{T}^{N},l^{N},r^{N})$
such that

(i):

the decorated PE metric of $(\mathcal{T}^{0},l^{0},r^{0})$ is $(dist_{S},r)$
and the decorated PE metric of $(\mathcal{T}^{N},l^{N},r^{N})$ is
$(\widetilde{dist}_{S},\widetilde{r})$,

(ii):

each $\mathcal{T}^{n}$ is a weighted Delaunay triangulation of the decorated
PE surface $(\mathcal{T}^{n},l^{n},r^{n})$,

(iii):

if $\mathcal{T}^{n}=\mathcal{T}^{n+1}$, then there is a discrete conformal
factor $u\in\mathbb{R}^{V}$ such that $(\mathcal{T}^{n},l^{n},r^{n})$ and
$(\mathcal{T}^{n+1},l^{n+1},r^{n+1})$ are related by (4) and (5),

(iv):

if $\mathcal{T}^{n}\neq\mathcal{T}^{n+1}$, then $\mathcal{T}^{n}$ and
$\mathcal{T}^{n+1}$ are two different weighted Delaunay triangulations of the
same decorated PE surface.

Definition 2.3 defines an equivalence relationship for decorated PE metrics on
a marked surface. The equivalence class of a decorated PE metric
$(dist_{S},r)$ on $(S,V)$ is also called as the discrete conformal class of
$(dist_{S},r)$ and denoted by $\mathcal{D}(dist_{S},r)$.

######  Lemma 2.4 ([2]).

The discrete conformal class $\mathcal{D}(dist_{S},r)$ of a decorated PE
metric $(dist_{S},r)$ on the marked surface $(S,V)$ is parameterized by
$\mathbb{R}^{V}=\\{u:V\rightarrow\mathbb{R}\\}$.

For simplicity, for any
$(\widetilde{dist}_{S},\widetilde{r})\in\mathcal{D}(dist_{S},r)$, we denote it
by $(dist_{S}(u),r(u))$ for some $u\in\mathbb{R}^{V}$. Set

$\mathcal{C}_{\mathcal{T}}(dist_{S},r)=\\{u\in\mathbb{R}^{V}|\ \mathcal{T}\
\text{is a weighted Delaunay triangulation of}\ (S,V,dist_{S}(u),r(u))\\}.$

For any decorated PE surface, there exists a unique complete hyperbolic
surface $\Sigma_{g}$, i.e., the hyperbolic surface induced by any triangular
refinement of its unique weighted Delaunay tessellation. It is homeomorphic to
$S\backslash V$ and called as the fundamental discrete conformal invariant of
the decorated PE metric $(dist_{S},r)$. The decoration of $\Sigma_{g}$ is
denoted by $\omega:=e^{h}$ and here the height $h$ is related to $u$ by
$dh_{i}=-du_{i}$. The canonical weighted Delaunay tessellation $\mathcal{T}$
of $\Sigma_{g}$ is denoted by $\mathcal{T}_{\Sigma_{g}}^{\omega}$. Bobenko-
Lutz [2] defined the following set

$\mathcal{D}_{\mathcal{T}}(\Sigma_{g})=\\{\omega\in\mathbb{R}_{>0}^{V}|\mathcal{T}\
\text{refines}\ \mathcal{T}_{\Sigma_{g}}^{\omega}\\}$

and proved the following proposition.

######  Proposition 2.5 ([2], Proposition 4.3).

Given a complete hyperbolic surface with ends $\Sigma_{g}$.

(1):

Each $\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$ is either empty or the
intersection of $\mathbb{R}^{V}_{>0}$ with a closed polyhedral cone.

(2):

There is only a finite number of geodesic tessellations
$\mathcal{T}_{1},...,\mathcal{T}_{N}$ of $\Sigma_{g}$ such that
$\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$ $(n=1,...,N)$ is non-empty. In
particular,
$\mathbb{R}^{V}_{>0}=\bigcup_{n=1}^{N}\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$.

Let $P$ be the polyhedral cusp corresponding to the triangulated surface
$(S,V,\mathcal{T})$ with fundamental discrete conformal invariant
$\Sigma_{g}$. The polyhedral cusp is convex if and only if $\mathcal{T}$ is a
weighted Delaunay triangulation. The set of all heights $h$ of convex
polyhedral cusps over the triangulated hyperbolic surface
$(\Sigma_{g},\mathcal{T})$ is denoted by
$\mathcal{P}_{\mathcal{T}}(\Sigma_{g})\subseteq\mathbb{R}^{V}$.

######  Proposition 2.6 ([2], Proposition 4.9).

Given a decorated PE metric $(dist_{S},r)$ on the marked surface $(S,V)$. Then
$\mathcal{C}_{\mathcal{T}}(dist_{S},r)$,
$\mathcal{P}_{\mathcal{T}}(\Sigma_{g})$ and
$\mathcal{D}_{\mathcal{T}}(\Sigma_{g})$ are homeomorphic.

Combining Proposition 2.5 and Proposition 2.6 gives the following result.

######  Lemma 2.7 ([2]).

The set

$J=\\{\mathcal{T}|\mathcal{C}_{\mathcal{T}}(dist_{S},r)\ \text{has non-empty
interior in}\ \mathbb{R}^{V}\\}$

is a finite set, $\mathbb{R}^{V}=\cup_{\mathcal{T}_{i}\in
J}\mathcal{C}_{\mathcal{T}_{i}}(dist_{S},r)$, and each
$\mathcal{C}_{\mathcal{T}_{i}}(dist_{S},r)$ is homeomorphic to a polyhedral
cone (with its apex removed) and its interior is homeomorphic to
$\mathbb{R}^{V}$.

###  2.2. A decorated triangle

Denote $r_{ij}$ as half of the distance of the two intersection points of the
face-circle $C_{ijk}$ and the edge $\\{ij\\}$. Denote $h_{ij}^{k}$ as the
signed distance of the center $c_{ijk}$ to the edge $\\{ij\\}$, which is
defined to be positive if the center is on the same side of the line
determined by $\\{ij\\}$ as the triangle $\\{ijk\\}$ and negative otherwise
(or zero if the center is on the line). Note that $h_{ij}^{k}$ is symmetric in
the indices $i$ and $j$. By Figure 3, we have

(7) $h_{ij}^{k}=r_{ij}\cot\alpha_{ij}^{k}.$

Since $r_{ij}>0$ and $\alpha_{ij}^{k}\in(0,\pi)$, if $h_{ij}^{k}<0$, then
$\alpha_{ij}^{k}\in(\frac{\pi}{2},\pi)$. The equality (7) implies that (2) is
equivalent to

(8) $h_{ij}^{k}+h_{ij}^{l}\geq 0$

for any adjacent triangles $\\{ijk\\}$ and $\\{ijl\\}$ sharing a common edge
$\\{ij\\}$. Therefore, the equality (8) also characterizes a weighted Delaunay
triangulation $\mathcal{T}$ for a decorated PE metric $(l,r)$ on $(S,V)$. Due
to this fact, the equality (8) is usually used to define the weighted Delaunay
triangulations of decorated PE surfaces. See [5, 8] and others for example.
Then $A_{i}^{jk}$ can be written as

(9) $A_{i}^{jk}=\frac{1}{2}l_{ij}h_{ij}^{k}+\frac{1}{2}l_{ki}h_{ki}^{j}.$

Since $h_{ij}^{k},h_{ki}^{j}$ are the signed distances, thus $A_{i}^{jk}$ is
an algebraic sum of the area of triangles, i.e. a signed area.

######  Lemma 2.8.

The area $A_{i}$ is independent of the choice of weighted Delaunay
triangulations of a decorated PE surface.

###### Proof.

Suppose a decorated quadrilateral $\\{ijlk\\}$ is in a face of the weighted
Delaunay tessellation of a decorated PE surface, then there exist two weighted
Delaunay triangulations $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ of the
decorated PE surface with an edge $\\{jk\\}$ in $\mathcal{T}_{1}$ flipped to
another edge $\\{il\\}$ in $\mathcal{T}_{2}$. Please refer to Figure 2. We
just need to prove the signed area $A^{jk}_{i}$ in $\mathcal{T}_{1}$ is equal
to the signed area $A_{i}^{kl}+A_{i}^{jl}$ in $\mathcal{T}_{2}$. In
$\mathcal{T}_{1}$, the signed area at the vertex $i$ in $\\{ijlk\\}$ is
$A_{i}^{jk}=\frac{1}{2}l_{ki}h_{ki}^{j}+\frac{1}{2}l_{ij}h_{ij}^{k}$. In
$\mathcal{T}_{2}$, the signed area at the vertex $i$ in $\\{ijlk\\}$ is

$\displaystyle A_{i}^{kl}+A_{i}^{jl}$
$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{l}+\frac{1}{2}l_{il}h_{il}^{k}+\frac{1}{2}l_{ij}h_{ij}^{l}+\frac{1}{2}l_{il}h_{il}^{j}$
$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{l}+\frac{1}{2}l_{ij}h_{ij}^{l}+\frac{1}{2}l_{il}(h_{il}^{k}+h_{il}^{j}).$

Since $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are two weighted Delaunay
triangulations of the same decorated PE metric on $(S,V)$, then
$h_{il}^{k}+h_{il}^{j}=0$ by (8). One can also refer to [2] (Proposition 3.4)
for this. Moreover, $h_{ki}^{l}=h_{ki}^{j}$ and $h_{ij}^{l}=h_{ij}^{k}$. Then
$A_{i}^{kl}+A_{i}^{jl}=A_{i}^{jk}$.

\begin{overpic}[scale={0.6}]{figure_2.pdf} \put(4.0,20.0){$i$}
\put(41.5,20.0){$l$} \put(23.0,41.0){$k$} \put(17.0,2.0){$j$}
\put(3.5,24.5){$r_{i}$} \put(14.0,17.0){$h_{ij}^{k}$}
\put(16.5,27.0){$h_{ki}^{j}$} \put(23.5,19.0){$c_{ijk}$}
\par\put(56.5,20.0){$i$} \put(94.0,20.0){$l$} \put(75.0,41.0){$k$}
\put(69.0,2.0){$j$} \put(56.0,24.5){$r_{i}$} \put(76.0,19.0){$c_{ijk}$}
\put(66.5,17.0){$h_{ij}^{l}$} \put(69.0,27.0){$h_{ki}^{l}$}
\put(76.0,21.0){$h_{il}^{j}$} \end{overpic} Figure 2. Weighted Delaunay
triangulation $\mathcal{T}_{1}$ (left) and weighted Delaunay triangulation
$\mathcal{T}_{2}$ (right).

Q.E.D.

Denote $c_{ij}$ as the center of the edge $\\{ij\\}$, which is obtained by
projecting the center $c_{ijk}$ to the line determined by $\\{ij\\}$. Denote
$d_{ij}$ as the signed distance of $c_{ij}$ to the vertex $i$, which is
positive if $c_{ij}$ is on the same side of $i$ as $j$ along the line
determined by $\\{ij\\}$ and negative otherwise (or zero if $c_{ij}$ is the
same as $i$). In general, $d_{ij}\neq d_{ji}$. Since the face-circle $C_{ijk}$
is orthogonal to the vertex-circle at the vertex $j$, we have

(10)
$r_{ijk}^{2}+r_{j}^{2}=d_{jk}^{2}+(h^{i}_{jk})^{2}=d_{ji}^{2}+(h^{k}_{ij})^{2}.$

Please refer to Figure 3 for this. Moreover, we have the following explicit
expressions of $d_{ij}$ and $h_{ij}^{k}$ due to Glickenstein [9], i.e.,

(11) $d_{ij}=\frac{r_{i}^{2}+r_{i}r_{j}I_{ij}}{l_{ij}},$

and

(12)
$h_{ij}^{k}=\frac{d_{ik}-d_{ij}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}},$

where $\theta^{i}_{jk}$ is the inner angle of the triangle $\\{ijk\\}$ at the
vertex $i$. The equality (11) implies that $d_{ij}\in\mathbb{R}$ is
independent of the existence of the center $c_{ijk}$. Since each pair of
vertex-circles is required to be separated, then $I>1$. This implies

$d_{rs}>0,\ \ \forall\\{r,s\\}\subseteq\\{i,j,k\\}.$

\begin{overpic}[scale={0.4}]{figure_1.pdf} \put(15.0,15.0){$i$}
\put(80.5,15.0){$j$} \put(48.5,74.0){$k$} \put(30.0,55.0){$\alpha_{ik}^{j}$}
\put(20.0,30.0){$\alpha_{ik}^{j}$} \put(37.5,27.0){$\alpha_{ij}^{k}$}
\put(27.0,12.5){$\alpha_{ij}^{k}$} \put(63.0,12.5){$\alpha_{ij}^{k}$}
\put(10.0,25.0){$r_{i}$} \put(46.0,25.0){$h_{ij}^{k}$}
\put(36.0,40.0){$h_{ki}^{j}$} \put(53.0,43.5){$h_{jk}^{i}$}
\put(31.0,21.0){$d_{ij}$} \put(45.0,36.0){$c_{ijk}$} \put(44.0,16.0){$c_{ij}$}
\put(34.0,16.0){$r_{ij}$} \put(26.0,42.0){$c_{ik}$} \put(66.0,46.0){$c_{jk}$}
\end{overpic} Figure 3. Dates for a decorated triangle $\\{ijk\\}\in F$

The following lemma gives some useful formulas.

######  Lemma 2.9 ([14, 24, 25]).

Let $\\{ijk\\}$ be a decorated triangle with the edge lengths
$l_{ij},l_{jk},l_{ki}$ defined by (5). If the decorated triangle $\\{ijk\\}$
is non-degenerate, then

(13) $\frac{\partial\theta_{jk}^{i}}{\partial
u_{j}}=\frac{\partial\theta_{ki}^{j}}{\partial
u_{i}}=\frac{h_{ij}^{k}}{l_{ij}},\ \ \ \frac{\partial\theta_{jk}^{i}}{\partial
u_{i}}=-\frac{\partial\theta_{jk}^{i}}{\partial
u_{j}}-\frac{\partial\theta_{jk}^{i}}{\partial u_{k}},$

where

(14)
$h_{ij}^{k}=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{2A_{ijk}l_{ij}}[\kappa_{k}^{2}(1-I_{k}^{2})+\kappa_{j}\kappa_{k}\gamma_{i}+\kappa_{i}\kappa_{k}\gamma_{j}]=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{2A_{ijk}l_{ij}}\kappa_{k}h_{k}$

with $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$,
$\gamma_{i}=I_{jk}+I_{ij}I_{ki}$, $\kappa_{i}:=r_{i}^{-1}$ and

(15) $\displaystyle
h_{i}=\kappa_{i}(1-I_{jk}^{2})+\kappa_{j}\gamma_{k}+\kappa_{k}\gamma_{j}.$

As a direct application of Lemma 2.9, we have the following result.

######  Lemma 2.10.

The area $A_{ijk}(u)$ of each decorated triangle $\\{ijk\\}\in F$ is an
analytic function with

(16) $\frac{\partial A_{ijk}}{\partial u_{i}}=A_{i}^{jk}.$

###### Proof.

By (14), we have

$h^{k}_{ij}=\frac{d_{ik}-d_{ij}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}},\quad
h_{ki}^{j}=\frac{d_{ij}-d_{ik}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}}.$

Direct calculations give

(17) $h_{ki}^{j}=d_{ij}\sin\theta_{jk}^{i}-h^{k}_{ij}\cos\theta_{jk}^{i}.$

Combining (5), (6) and (11), it is easy to check that

(18) $\frac{\partial l_{ij}}{\partial u_{i}}=d_{ij}.$

Differentiating $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$ with
respect to $u_{i}$ gives

$\displaystyle\frac{\partial A_{ijk}}{\partial u_{i}}$
$\displaystyle=\frac{1}{2}\frac{\partial l_{ij}}{\partial
u_{i}}l_{jk}\sin\theta_{ki}^{j}+\frac{1}{2}l_{ij}l_{jk}\cos\theta_{ki}^{j}\frac{\partial\theta_{ki}^{j}}{\partial
u_{i}}$
$\displaystyle=\frac{1}{2}d_{ij}l_{jk}\sin\theta_{ki}^{j}+\frac{1}{2}l_{ij}l_{jk}\cos\theta_{ki}^{j}\frac{h^{k}_{ij}}{l_{ij}}$
$\displaystyle=\frac{1}{2}d_{ij}l_{ki}\sin\theta_{jk}^{i}+\frac{1}{2}l_{jk}\cos\theta_{ki}^{j}h^{k}_{ij}$
$\displaystyle=\frac{1}{2}d_{ij}l_{ki}\sin\theta_{jk}^{i}+\frac{1}{2}(l_{ij}-l_{ki}\cos\theta_{jk}^{i})h^{k}_{ij}$
$\displaystyle=\frac{1}{2}l_{ki}(d_{ij}\sin\theta_{jk}^{i}-h^{k}_{ij}\cos\theta_{jk}^{i})+\frac{1}{2}l_{ij}h^{k}_{ij}$
$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{j}+\frac{1}{2}l_{ij}h_{ij}^{k}$
$\displaystyle=A_{i}^{jk},$

where the second equality uses (18) and (13), the third equality uses the sine
laws and the penultimate line uses (17). Q.E.D.

######  Remark 2.11.

One can refer to Glickenstein [9] for a nice geometric explanation of the
result in Lemma 2.10.

###  2.3. The extended energy function and the extended area function

There exists a geometric relationship between the decorated triangle
$\\{ijk\\}$ and the geometry of hyperbolic polyhedra in $3$-dimensional
hyperbolic space. Specially, there is a generalized hyperbolic tetrahedra in
$\mathbb{H}^{3}$ with one ideal vertex and three hyper-ideal vertices
corresponding to a decorated triangle $\\{ijk\\}$. Please refer to [2] for
more details on this fact. Springborn [22] found the following explicit
formula for the truncated volume $\mathrm{Vol}(ijk)$ of this generalized
hyperbolic tetrahedra

(19) $\displaystyle 2\mathrm{Vol}(ijk)=$
$\displaystyle\mathbb{L}(\theta_{jk}^{i})+\mathbb{L}(\theta_{ki}^{j})+\mathbb{L}(\theta_{ij}^{k})$
$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{ki}^{j}+\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})+\mathbb{L}(\frac{\pi+\alpha_{ki}^{j}-\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})$
$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{ki}^{j}+\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})+\mathbb{L}(\frac{\pi-\alpha_{ki}^{j}-\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})$
$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}+\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}-\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})$
$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}+\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}-\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})$
$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}+\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}-\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})$
$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}+\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}-\alpha_{ki}^{j}-\theta_{ij}^{k}}{2}),$

where

(20) $\mathbb{L}(x)=-\int_{0}^{x}\log|2\sin(t)|dt$

is Milnor’s Lobachevsky function. Milnor’s Lobachevsky function is bounded,
odd, $\pi$-periodic and smooth except at integer multiples of $\pi$. Please
refer to [20, 21] for more information on Milnor’s Lobachevsky function
$\mathbb{L}(x)$.

Set

(21) $\displaystyle F_{ijk}(u_{i},u_{j},u_{k})=$
$\displaystyle-2\mathrm{Vol}(ijk)+\theta_{jk}^{i}u_{i}+\theta_{ki}^{j}u_{j}+\theta_{ij}^{k}u_{k}$
$\displaystyle+(\frac{\pi}{2}-\alpha_{ij}^{k})\lambda_{ij}+(\frac{\pi}{2}-\alpha_{ki}^{j})\lambda_{ki}+(\frac{\pi}{2}-\alpha_{jk}^{i})\lambda_{jk},$

where $\cosh\lambda_{ij}=I_{ij}$. Then $\nabla
F_{ijk}=(\theta_{jk}^{i},\theta_{ki}^{j},\theta_{ij}^{k})$ and

(22) $F_{ijk}((u_{i},u_{j},u_{k})+c(1,1,1))=F_{ijk}(u_{i},u_{j},u_{k})+c\pi$

for $c\in\mathbb{R}$. Furthermore, on a decorated PE surface $(S,V,l,r)$ with
a weighted Delaunay triangulation $\mathcal{T}$, Bobenko-Lutz [2] defined the
following function

(23) $\mathcal{H}_{\mathcal{T}}(u)=\sum_{\\{ijk\\}\in
F}F_{ijk}(u_{i},u_{j},u_{k})=-2\mathrm{Vol}(P_{h})+\sum_{i\in
V}\theta_{i}u_{i}+\sum_{\\{ij\\}\in
E_{\mathcal{T}}}(\pi-\alpha_{ij})\lambda_{ij},$

where $P_{h}$ is the convex polyhedral cusp defined by the heights
$h\in\mathbb{R}^{V}$, $\theta_{i}=\sum_{\\{ijk\\}\in
F_{\mathcal{T}}}\theta^{i}_{jk}$ and
$\alpha_{ij}=\alpha_{ij}^{k}+\alpha_{ij}^{l}$. Note that the function
$\mathcal{H}_{\mathcal{T}}(u)$ defined by (23) differs from its original
definition in [2] (Equation 4-9) by some constant. By (22), for
$c\in\mathbb{R}$, we have

$\mathcal{H}_{\mathcal{T}}(u+c\mathbf{1})=\mathcal{H}_{\mathcal{T}}(u)+c|F|\pi.$

Using the function $\mathcal{H}_{\mathcal{T}}$, we define the following energy
function

$\mathcal{E}_{\mathcal{T}}(u)=-\mathcal{H}_{\mathcal{T}}(u)+2\pi\sum_{i\in
V}u_{i},$

which is well-defined on $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ with
$\nabla_{u_{i}}\mathcal{E}_{\mathcal{T}}=2\pi-\sum_{\\{ijk\\}\in
F_{\mathcal{T}}}\theta^{i}_{jk}=W_{i}$. Moreover, for $c\in\mathbb{R}$, we
have

(24) $\displaystyle\mathcal{E}_{\mathcal{T}}(u+c\mathbf{1})=$
$\displaystyle-\mathcal{H}_{\mathcal{T}}(u+c\mathbf{1})+2\pi\sum_{i\in
V}(u_{i}+c)$ $\displaystyle=$
$\displaystyle-\mathcal{H}_{\mathcal{T}}(u)-c|F|\pi+2\pi\sum_{i\in
V}u_{i}+2c|V|\pi$ $\displaystyle=$
$\displaystyle\mathcal{E}_{\mathcal{T}}(u)+2c\pi\chi(S),$

where $2|V|-|F|=2\chi(S)$ is used in the last line.

######  Theorem 2.12 ([2], Proposition 4.13).

For a discrete conformal factor $u\in\mathbb{R}^{V}$, let $\mathcal{T}$ be a
weighted Delaunay triangulation of the decorated PE surface
$(S,V,dist_{S}(u),r(u))$. The map

(25) $\displaystyle\mathcal{H}:\ \mathbb{R}^{V}$
$\displaystyle\rightarrow\mathbb{R},$ $\displaystyle u$
$\displaystyle\mapsto\mathcal{H}_{\mathcal{T}}(u)$

is well-defined, concave, and twice continuously differentiable over
$\mathbb{R}^{V}$.

Therefore, the function $\mathcal{E}_{\mathcal{T}}(u)$ defined on
$\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ can be extended to be

(26) $\mathcal{E}(u)=-\mathcal{H}(u)+2\pi\sum_{i\in
V}u_{i}=-\sum_{\\{ijk\\}\in F}F_{ijk}(u_{i},u_{j},u_{k})+2\pi\sum_{i\in
V}u_{i}$

defined on $\mathbb{R}^{V}$.

######  Definition 2.13.

Suppose $(S,V,\mathcal{T})$ is a triangulated surface with a decorated PE
metric $(l,r)$. The area function $A^{\mathcal{T}}_{tot}$ on
$(S,V,\mathcal{T})$ is defined to be

$A^{\mathcal{T}}_{tot}:\
\mathcal{C}_{\mathcal{T}}(dist_{S},r)\rightarrow\mathbb{R},\\\ \quad\
A^{\mathcal{T}}_{tot}(u)=\sum_{\\{ijk\\}\in F}A_{ijk}(u).$

By Lemma 2.10, we have the following result.

######  Corollary 2.14.

The function $A^{\mathcal{T}}_{tot}$ is an analytic function with

(27) $\frac{\partial A^{\mathcal{T}}_{tot}}{\partial u_{i}}=A_{i}.$

Lemma 2.8 and Corollary 2.14 imply the following result, which shows the
function $A^{\mathcal{T}}_{tot}$ defined on
$\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ can be extended.

######  Theorem 2.15.

For a discrete conformal factor $u\in\mathbb{R}^{V}$, let $\mathcal{T}$ be a
weighted Delaunay triangulation of the decorated PE surface
$(S,V,dist_{S}(u),r(u))$. The map

(28) $\displaystyle A_{tot}:\ \mathbb{R}^{V}$
$\displaystyle\rightarrow\mathbb{R},$ $\displaystyle u$ $\displaystyle\mapsto
A^{\mathcal{T}}_{tot}(u)$

is well-defined and once differentiable.

###### Proof.

By Corollary 2.14, the function $A_{tot}$ is once differentiable in the
interior of any $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$. At the boundary of
$\mathcal{C}_{\mathcal{T}}(dist_{S},r)$, the weighted triangulations induce
the same weighted Delaunay tessellation. The conclusion follows from Lemma
2.8. Q.E.D.

##  3\. The proof of Theorem 1.5

###  3.1. Variational principles with constraints

In this subsection, we translate Theorem 1.5 into an optimization problem with
inequality constraints by variational principles, which involves the function
$\mathcal{E}$ defined by (26).

######  Proposition 3.1.

The set

$\mathcal{A}=\\{u\in\mathbb{R}^{V}|A_{tot}(u)\leq 1\\}.$

is an unbounded closed subset of $\mathbb{R}^{V}$.

###### Proof.

By Theorem 2.15, the set $\mathcal{A}$ is a closed subset of $\mathbb{R}^{V}$.
Since $A_{ijk}((u_{i},u_{j},u_{k})+(c,c,c))=e^{2c}A_{ijk}(u_{i},u_{j},u_{k})$,
thus $A_{tot}(u+c\mathbf{1})=e^{2c}A_{tot}(u)$. Then
$A_{tot}(u+c\mathbf{1})=e^{2c}A_{tot}(u)\leq 1$ is equivalent to
$c\leq-\frac{1}{2}\log A_{tot}(u)$. This implies that the ray
$\\{u+c\mathbf{1}|c\leq-\frac{1}{2}\log A_{tot}(u)\\}$ stays in the set
$\mathcal{A}$. Hence the set $\mathcal{A}$ is unbounded. Q.E.D.

According to Proposition 3.1, we have following result.

######  Lemma 3.2.

If $\chi(S)<0$ and the function $\mathcal{E}(u)$ attains a minimum in the set
$\mathcal{A}$, then the minimum value point of $\mathcal{E}(u)$ lies at the
boundary of $\mathcal{A}$, i.e.,

$\partial\mathcal{A}=\\{u\in\mathbb{R}^{V}|A_{tot}(u)=1\\}.$

Furthermore, there exists a decorated PE metric with constant discrete
Gaussian curvature $K$ in the discrete conformal class.

###### Proof.

Suppose the function $\mathcal{E}(u)$ attains a minimum at $u\in\mathcal{A}$.
Taking $c_{0}=-\frac{1}{2}\log A_{tot}(u)$, then $c_{0}\geq 0$ by
$A_{tot}(u)\leq 1$. By the proof of Proposition 3.1,
$u+c_{0}\mathbf{1}\in\mathcal{A}$. Hence, by the additive property of the
function $\mathcal{E}$ in (24), we have

$\mathcal{E}(u)\leq\mathcal{E}(u+c_{0}\mathbf{1})=\mathcal{E}(u)+2c_{0}\pi\chi(S).$

This implies $c_{0}\leq 0$ by $\chi(S)<0$. Then $c_{0}=0$ and $A_{tot}(u)=1$.
Therefore, the minimum value point of $\mathcal{E}(u)$ lies in the set
$\partial\mathcal{A}=\\{u\in\mathbb{R}^{V}|A_{tot}(u)=1\\}$. The conclusion
follows from the following claim.
Claim : Up to scaling, the decorated PE metrics with constant discrete
Gaussian curvature $K$ in the discrete conformal class are in one-to-one
correspondence with the critical points of the function $\mathcal{E}(u)$ under
the constraint $A_{tot}(u)=1$.

We use the method of Lagrange multipliers to prove this claim. Set

$G(u,\mu)=\mathcal{E}(u)-\mu(A_{tot}(u)-1),$

where $\mu\in\mathbb{R}$ is a Lagrange multiplier. If $u$ is a critical point
of the function $\mathcal{E}$ under the constraint $A_{tot}(u)=1$, then by
(27) and the fact $\nabla_{u_{i}}\mathcal{E}=W_{i}$, we have

$0=\frac{\partial G(u,\mu)}{\partial
u_{i}}=\frac{\partial\mathcal{E}(u)}{\partial u_{i}}-\mu\frac{\partial
A_{tot}(u)}{\partial u_{i}}=W_{i}-\mu A_{i}.$

This implies

$W_{i}=\mu A_{i}.$

Since the anger defect $W$ defined by (1) satisfies the following discrete
Gauss-Bonnet formula

$\sum_{i\in V}W_{i}=2\pi\chi(S),$

we have

$2\pi\chi(S)=\sum_{i\in V}W_{i}=\mu\sum_{i\in V}A_{i}=\mu A_{tot}=\mu.$

under the constraint $A_{tot}(u)=1$. Therefore, the discrete Gaussian
curvature

$K_{i}=\frac{W_{i}}{A_{i}}=2\pi\chi(S)$

for any $i\in V$. Q.E.D.

###  3.2. Reduction to Theorem 3.4

By Lemma 3.2, we just need to prove that the function $\mathcal{E}(u)$ attains
the minimum in the set $\mathcal{A}$. Recall the following classical result
from calculus.

######  Theorem 3.3.

Let $A\subseteq\mathbb{R}^{m}$ be a closed set and $f:A\rightarrow\mathbb{R}$
be a continuous function. If every unbounded sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ in $A$ has a subsequence
$\\{u_{n_{k}}\\}_{k\in\mathbb{N}}$ such that
$\lim_{k\rightarrow+\infty}f(u_{n_{k}})=+\infty$, then $f$ attains a minimum
in $A$.

One can refer to [16] (Section 4.1) for a proof of Theorem 3.3. The majority
of the conditions in Theorem 3.3 is satisfied, including the set $\mathcal{A}$
is a closed subset of $\mathbb{R}^{V}$ by Proposition 3.1 and the function
$\mathcal{E}$ is continuous by Theorem 2.12. To prove Theorem 1.5, we just
need to prove the following theorem.

######  Theorem 3.4.

If $\chi(S)<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{A}$, then there exists a subsequence
$\\{u_{n_{k}}\\}_{k\in\mathbb{N}}$ of $\\{u_{n}\\}_{n\in\mathbb{N}}$ such that
$\lim_{k\rightarrow+\infty}\mathcal{E}(u_{n_{k}})=+\infty$.

###  3.3. Behaviour of sequences of discrete conformal factors

Let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be an unbounded sequence in
$\mathbb{R}^{V}$. Denote its coordinate sequence at $j\in V$ by
$\\{u_{j,n}\\}_{n\in\mathbb{N}}$. Motivated by [17], we call the sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ with the following properties as a “good”
sequence.

(1):

It lies in one cell $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ of
$\mathbb{R}^{V}$;

(2):

There exists a vertex $i^{*}\in V$ such that $u_{i^{*},n}\leq u_{j,n}$ for all
$j\in V$ and $n\in\mathbb{N}$;

(3):

Each coordinate sequence $\\{u_{j,n}\\}_{n\in\mathbb{N}}$ either converges,
diverges properly to $+\infty$, or diverges properly to $-\infty$;

(4):

For any $j\in V$, the sequence $\\{u_{j,n}-u_{i^{*},n}\\}_{n\in\mathbb{N}}$
either converges or diverges properly to $+\infty$.

By Lemma 2.7, it is obvious that every sequence of discrete conformal factors
in $\mathbb{R}^{V}$ possesses a “good” subsequence. Hence, the “good” sequence
could be chosen without loss of generality.

In the following arguments, we use the following notations

(29) $l^{n}_{ij}=\sqrt{r^{2}_{i,n}+r^{2}_{j,n}+2I_{ij}r_{i,n}r_{j,n}},$ (30)
$r_{i,n}=e^{u_{i,n}}r_{i},$ (31)
$(l^{n}_{ij})^{2}=(e^{2u_{i,n}}-e^{u_{i,n}+u_{j,n}})r^{2}_{i}+(e^{2u_{j,n}}-e^{u_{i,n}+u_{j,n}})r^{2}_{j}+e^{{u_{i,n}+u_{j,n}}}l_{ij}^{2}.$

For a decorated triangle $\\{ijk\\}\in F$ in $(S,V,\mathcal{T})$, set

(32)
$\mathcal{C}_{ijk}=\\{(u_{i},u_{j},u_{k})\in\mathbb{R}^{3}|u\in\mathcal{C}_{\mathcal{T}}(dist_{S},r)\\}.$

Let $(u_{i,n},u_{j,n},u_{k,n})_{n\in\mathbb{N}}$ be a coordinate sequence in
$\mathcal{C}_{ijk}$. Then the edge lengths $l^{n}_{ij},l^{n}_{jk},l^{n}_{ki}$
satisfy the triangle inequalities for all $n\in\mathbb{N}$.

######  Lemma 3.5.

There exists no sequence in $\mathcal{C}_{ijk}$ such that as
$n\rightarrow+\infty$,

$u_{r,n}\rightarrow+\infty,\quad u_{s,n}\rightarrow+\infty,\quad u_{t,n}\leq
C,$

where $\\{r,s,t\\}=\\{i,j,k\\}$ and $C$ is a constant.

###### Proof.

Without loss of generality, we assume $\lim u_{i,n}=+\infty$, $\lim
u_{j,n}=+\infty$ and the sequence $u_{k,n}\leq C_{1}$. The equality (30)
implies $\lim r_{i,n}=+\infty$, $\lim r_{j,n}=+\infty$ and the sequence
$r_{k,n}\leq C_{2}$. Here $C_{1},C_{2}$ are constants. By (29), we have

$\displaystyle(l^{n}_{jk}+l^{n}_{ki})^{2}=$ $\displaystyle
r^{2}_{i,n}+r^{2}_{j,n}+2r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n}+2I_{ki}r_{k,n}r_{i,n}$
$\displaystyle+2\sqrt{(r^{2}_{j,n}+r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n})(r^{2}_{k,n}+r^{2}_{i,n}+2I_{ki}r_{k,n}r_{i,n})}.$

Note that $I_{ij}>1$, then

$\lim\frac{r^{2}_{k,n}+I_{jk}r_{j,n}r_{k,n}+I_{ki}r_{k,n}r_{i,n}+\sqrt{(r^{2}_{j,n}+r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n})(r^{2}_{k,n}+r^{2}_{i,n}+2I_{ki}r_{k,n}r_{i,n})}}{I_{ij}r_{i,n}r_{j,n}}<1.$

Therefore, there exists $n\in\mathbb{N}$ such that
$(l^{n})^{2}_{ij}=r^{2}_{i,n}+r^{2}_{j,n}+2I_{ij}r_{i,n}r_{j,n}>(l^{n}_{jk}+l^{n}_{ki})^{2}$,
i.e., $l^{n}_{ij}>l^{n}_{jk}+l^{n}_{ki}$. This contradicts the triangle
inequality $l^{n}_{ij}<l^{n}_{jk}+l^{n}_{ki}$. Q.E.D.

Combining Lemma 3.5 and the connectivity of the triangulation $\mathcal{T}$,
we have the following result.

######  Corollary 3.6.

For a discrete conformal factor $u\in\mathbb{R}^{V}$, let $\mathcal{T}$ be a
weighted Delaunay triangulation of the decorated PE surface
$(S,V,dist_{S}(u),r(u))$. For any decorated triangle $\\{ijk\\}\in F$ in
$\mathcal{T}$, at least two of the three sequences
$(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$,
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$,
$(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge.

To characterize the function $F_{ijk}(u_{i},u_{j},u_{k})$ in (21), we need the
following lemmas.

######  Lemma 3.7.

Assume that the sequence $(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to
$+\infty$ and the sequences $(u_{j,n})_{n\in\mathbb{N}}$ and
$(u_{k,n})_{n\in\mathbb{N}}$ converge. Then the sequence
$(\theta^{i,n}_{jk})_{n\in\mathbb{N}}$ converges to zero. Furthermore, if the
sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants, then

(1):

the sequences $(h_{jk}^{i,n})_{n\in\mathbb{N}}$,
$(h_{ki}^{j,n})_{n\in\mathbb{N}}$ and $(h_{ij}^{k,n})_{n\in\mathbb{N}}$
converge;

(2):

the sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$,
$(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge.

###### Proof.

By the assumption, we have $\lim r_{i,n}=+\infty$, $\lim r_{j,n}=c_{1}$ and
$\lim r_{k,n}=c_{2}$, where $c_{1},c_{2}$ are positive constants. The equality
(29) implies

(33) $\lim\frac{l_{ij}^{n}}{r_{i,n}}=1,\ \lim\frac{l_{ki}^{n}}{r_{i,n}}=1,\
\lim l_{jk}^{n}=c_{3},$

where $c_{3}$ is a positive constant. By the cosine law, we have

$\lim\cos\theta^{i,n}_{jk}=\lim\frac{-(l_{jk}^{n})^{2}+(l_{ij}^{n})^{2}+(l_{ki}^{n})^{2}}{2l_{ij}^{n}l_{ki}^{n}}=1.$

This implies $\lim\theta^{i,n}_{jk}=0$.

Suppose the sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants. Then

(34)
$\lim\frac{A_{ijk}^{n}}{r_{i,n}}=\lim\frac{l^{n}_{ij}l^{n}_{jk}\sin\theta^{j,n}_{ki}}{2r_{i,n}}=c_{4}$

for some constant $c_{4}>0$.

(1): Since $\kappa_{i}=\frac{1}{r_{i}}$, then $\lim\kappa_{i,n}=0$,
$\lim\kappa_{j,n}=\frac{1}{c_{1}}$ and $\lim\kappa_{k,n}=\frac{1}{c_{2}}$. By
(15), we have

$\displaystyle\lim h_{i,n}=$
$\displaystyle\lim(\kappa_{i,n}(1-I_{jk}^{2})+\kappa_{j,n}\gamma_{k}+\kappa_{k,n}\gamma_{j})=c_{5}>0,$
$\displaystyle\lim h_{j,n}=$
$\displaystyle\lim(\kappa_{j,n}(1-I_{ki}^{2})+\kappa_{i,n}\gamma_{k}+\kappa_{k,n}\gamma_{i})=c_{6},$
$\displaystyle\lim h_{k,n}=$
$\displaystyle\lim(\kappa_{k,n}(1-I_{ij}^{2})+\kappa_{i,n}\gamma_{j}+\kappa_{j,n}\gamma_{i})=c_{7},$

where $c_{5},c_{6},c_{7}$ are constants. Note that $c_{6},c_{7}$ may be non-
positive. The equalities (14) and (34) imply

$\displaystyle\lim h_{jk}^{i,n}=$
$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{jk}}\kappa_{i,n}h_{i,n}=\frac{c_{1}^{2}c^{2}_{2}c_{5}}{2c_{3}c_{4}}>0,$
$\displaystyle\lim h_{ki}^{j,n}=$
$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{ki}}\kappa_{j,n}h_{j,n}=\frac{c_{1}c^{2}_{2}c_{6}}{2c_{4}},$
$\displaystyle\lim h_{ij}^{k,n}=$
$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{ij}}\kappa_{k,n}h_{k,n}=\frac{c_{1}^{2}c_{2}c_{7}}{2c_{4}}.$

Hence the sequences $(h_{jk}^{i,n})_{n\in\mathbb{N}}$,
$(h_{ki}^{j,n})_{n\in\mathbb{N}}$ and $(h_{ij}^{k,n})_{n\in\mathbb{N}}$
converge.

(2): The equality (11) implies

(35) $\lim
d^{n}_{jk}=\lim\frac{r_{j,n}^{2}+r_{j,n}r_{k,n}I_{jk}}{l^{n}_{jk}}=\frac{c_{1}^{2}+c_{1}c_{2}I_{jk}}{c_{3}}>0.$

By (10), we have

$\lim(r^{n}_{ijk})^{2}=\lim[(d^{n}_{jk})^{2}+(h^{i,n}_{jk})^{2}-r_{j,n}^{2}]=c_{8}.$

where $c_{8}$ is a constant. Note that
$h^{i}_{jk}=r_{ijk}\cos\alpha_{jk}^{i}$. Hence,

$\displaystyle\lim\cos\alpha_{jk}^{i,n}=$
$\displaystyle\lim\frac{h_{jk}^{i,n}}{r^{n}_{ijk}}=\frac{c_{1}^{2}c^{2}_{2}c_{5}}{2c_{3}c_{4}\sqrt{c_{8}}}>0,$
$\displaystyle\lim\cos\alpha_{ki}^{j,n}=$
$\displaystyle\lim\frac{h_{ki}^{j,n}}{r^{n}_{ijk}}=\frac{c_{1}c^{2}_{2}c_{6}}{2c_{4}\sqrt{c_{8}}},$
$\displaystyle\lim\cos\alpha_{ij}^{k,n}=$
$\displaystyle\lim\frac{h_{ij}^{k,n}}{r^{n}_{ijk}}=\frac{c_{1}^{2}c_{2}c_{7}}{2c_{4}\sqrt{c_{8}}}.$

Then the sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$,
$(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge. Q.E.D.

######  Lemma 3.8.

Assume that the sequence $(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to
$+\infty$ and the sequences $(u_{j,n})_{n\in\mathbb{N}}$ and
$(u_{k,n})_{n\in\mathbb{N}}$ converge. If the sequence
$(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ converge to zero, then

$\lim h^{i,n}_{jk}=+\infty,\ \lim h^{j,n}_{ki}=+\infty,\ \lim
h^{k,n}_{ij}=-\infty.$

###### Proof.

Lemma 3.7 shows that $\lim\theta^{i,n}_{jk}=0$, thus
$\lim(\theta^{j,n}_{ki}+\theta^{k,n}_{ij})=\pi$. Since
$\lim\theta^{j,n}_{ki}=0$, then $\lim\theta^{k,n}_{ij}=\pi$. Then

(36)
$\lim\frac{A_{ijk}^{n}}{r_{i,n}}=\lim\frac{l^{n}_{ij}l^{n}_{jk}\sin\theta^{j,n}_{ki}}{r_{i,n}}=0.$

By the proof of Lemma 3.7, we have

$\lim
h_{jk}^{i,n}=\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{jk}}\kappa_{i,n}h_{i,n}=\lim\frac{r_{i,n}^{2}c_{1}^{2}c_{2}^{2}}{2A^{n}_{ijk}c_{3}}\cdot\frac{1}{r_{i,n}}c_{5}=+\infty,$

where (36) is used and $c_{1},c_{2},c_{3},c_{5}$ are positive constants.
Similar to (35), we have

$\displaystyle\lim d^{n}_{ji}=$
$\displaystyle\lim\frac{r_{j,n}^{2}+r_{i,n}r_{j,n}I_{ij}}{l^{n}_{ij}}=c_{9},$
$\displaystyle\lim d^{n}_{ki}=$
$\displaystyle\lim\frac{r_{k,n}^{2}+r_{i,n}r_{k,n}I_{ki}}{l^{n}_{ki}}=c_{10}.$

Here $c_{9},c_{10}$ are positive constants. By (10), we have

$\displaystyle(r^{n}_{ijk})^{2}$
$\displaystyle=(d^{n}_{jk})^{2}+(h^{i,n}_{jk})^{2}-r_{j,n}^{2}$
$\displaystyle=(d^{n}_{ji})^{2}+(h^{k,n}_{ij})^{2}-r_{j,n}^{2}$
$\displaystyle=(d^{n}_{ki})^{2}+(h^{j,n}_{ki})^{2}-r_{k,n}^{2}.$

This implies $\lim r^{n}_{ijk}=+\infty$, $\lim(h^{k,n}_{ij})^{2}=+\infty$ and
$\lim(h^{j,n}_{ki})^{2}=+\infty$. Therefore, we have the following four cases

$(i)$:

$\lim h^{k,n}_{ij}=+\infty$ and $\lim h^{j,n}_{ki}=+\infty$;

$(ii)$:

$\lim h^{k,n}_{ij}=-\infty$ and $\lim h^{j,n}_{ki}=-\infty$;

$(iii)$:

$\lim h^{k,n}_{ij}=+\infty$ and $\lim h^{j,n}_{ki}=-\infty$;

$(iv)$:

$\lim h^{k,n}_{ij}=-\infty$ and $\lim h^{j,n}_{ki}=+\infty$.

For the case $(i)$, since $\lim h_{jk}^{i,n}>0$, $\lim h^{k,n}_{ij}>0$ and
$\lim h^{j,n}_{ki}>0$. This implies that the center $c_{ijk}$ of the face-
circle $C_{ijk}$ lies in the interior of the triangle $\\{ijk\\}$ by the
definition of $h_{jk}^{i},h^{k}_{ij},h^{j}_{ki}$. However, in this case, $\lim
h_{jk}^{i,n},\lim h^{k,n}_{ij},\lim h^{j,n}_{ki}$ are bounded. This is a
contradiction.

\begin{overpic}[scale={0.8}]{figure_4.pdf} \put(48.0,11.0){$i$}
\put(96.0,-2.0){$j$} \put(82.0,-1.0){$k$} \end{overpic} Figure 4. Domain of
the center $c_{ijk}$ in the decorated triangle on surface

Both the cases $(ii)$ and $(iii)$ imply $d_{kj}<0$. This contradicts with the
fact that $d_{rs}>0$ for any $\\{r,s\\}\subseteq\\{i,j,k\\}$. Indeed, the
center $c_{ijk}$ lies in the red region in Figure 4 in the case $(ii)$ and
lies in the blue region in Figure 4 in the case $(iii)$. By projecting the
center $c_{ijk}$ to the line determined by $\\{jk\\}$, we have $d_{kj}<0$.
This completes the proof. Q.E.D.

######  Remark 3.9.

Similar to the proof of Lemma 3.8, if the sequence
$(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converges to zero, then $\lim
h^{i,n}_{jk}=+\infty,\ \lim h^{j,n}_{ki}=-\infty,\ \lim h^{k,n}_{ij}=+\infty.$

Consider a star-shaped $s$-sided polygon in the marked surface with boundary
vertices $1,\cdots,s$ ordered cyclically ($v_{s+1}=v_{1}$). Please refer to
Figure 5. Let $i\in V$ be a vertex such that the sequence
$(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to $+\infty$ and the sequences
$(u_{j,n})_{n\in\mathbb{N}}$ converge for $j\sim i$.

\begin{overpic}[scale={0.5}]{figure_5.pdf} \put(74.0,90.0){$j+1$}
\put(92.0,44.0){$j$} \put(87.0,14.0){$j-1$} \put(50.0,38.0){$i$}
\put(78.0,16.0){$0$} \put(82.0,21.0){$\pi$} \put(85.0,39.0){$0$}
\put(85.0,46.0){$\pi$} \put(70.0,82.0){$0$} \put(64.0,86.0){$\pi$}
\put(38.0,92.0){$0$} \put(33.0,90.0){$\pi$} \put(35.0,100.0){$s$}
\put(6.0,55.0){$0$} \put(5.0,48.0){$\pi$} \put(-4.0,50.0){$1$}
\put(18.0,18.0){$0$} \put(23.0,14.0){$\pi$} \put(16.0,8.0){$2$}
\put(53.0,4.0){$0$} \put(59.0,5.0){$\pi$} \end{overpic} Figure 5. A star
triangulation of a star-shaped $s$-sided polygon

######  Lemma 3.10.

The sequences of inner angles at the boundary vertices in the triangles of a
star-shaped polygon converge to non-zero constants.

###### Proof.

As $\lim u_{i,n}=+\infty$ and $\lim u_{j,n}=C$ for $j\sim i$. By Lemma 3.7,
for any $j=1,...,s$, we have $\lim\theta^{i,n}_{j-1,j}=0$ and hence
$\lim(\theta^{j-1,n}_{i,j}+\theta^{j,n}_{i,j-1})=\pi$. We prove the result by
contradiction. Without loss of generality, we assume
$\lim\theta^{j-1,n}_{i,j}=\pi$ and $\lim\theta^{j,n}_{i,j-1}=0$ in the
triangle $\\{i,j-1,j\\}$. Then for $n$ large enough, we have

$l^{n}_{i,j-1}<l^{n}_{i,j}.$

By Lemma 3.8, we have $\lim h^{j,n}_{i,j-1}=+\infty,\ \lim
h^{j-1,n}_{i,j}=-\infty,\ \lim h^{i,n}_{j-1,j}=+\infty$. Since the edge
$\\{i,j\\}$ is weighted Delaunay, thus by (8), we have

$h^{j-1,n}_{i,j}+h^{j+1,n}_{i,j}\geq 0.$

This implies $\lim h^{j+1,n}_{i,j}=+\infty$.

In the triangle $\\{i,j,j+1\\}$, suppose the sequences
$(\theta^{j,n}_{i,j+1})_{n\in\mathbb{N}}$ and
$(\theta^{j+1,n}_{i,j})_{n\in\mathbb{N}}$ converge to non-zero constants. By
Lemma 3.7, the sequences $(h_{i,j}^{j+1,n})_{n\in\mathbb{N}}$ and
$(h_{i,j+1}^{j,n})_{n\in\mathbb{N}}$ converge. This contradicts $\lim
h^{j+1,n}_{i,j}=+\infty$. Hence the sequences
$(\theta^{j,n}_{i,j+1})_{n\in\mathbb{N}}$ or
$(\theta^{j+1,n}_{i,j})_{n\in\mathbb{N}}$ converge to zero. By Lemma 3.8 and
Remark 3.9, we have $\lim\theta^{j,n}_{i,j+1}=\pi$,
$\lim\theta^{j+1,n}_{i,j}=0$ and $\lim h^{j,n}_{i,j+1}=-\infty$. Then for $n$
large enough, we have

$l^{n}_{i,j}<l^{n}_{i,j+1}.$

Please refer to Figure 5. By induction, for $n$ large enough, we have

$l^{n}_{i,j-1}<l^{n}_{i,j}<l^{n}_{i,j+1}<l^{n}_{i,j+2}<...<l^{n}_{i,j-1}.$

This is a contradiction. Q.E.D.

Combining (21), Lemma 3.7 and Lemma 3.10, we have the following result.

######  Corollary 3.11.

Assume that the sequence $(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to
$+\infty$ and the sequences $(u_{j,n})_{n\in\mathbb{N}}$ and
$(u_{k,n})_{n\in\mathbb{N}}$ converge. Then the sequence
$(F_{ijk}(u_{i,n},u_{j,n},u_{k,n}))_{n\in\mathbb{N}}$ converges.

###### Proof.

By the definition of $F_{ijk}(u_{i},u_{j},u_{k})$ in (21), we have

$\displaystyle F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=$
$\displaystyle-2\mathrm{Vol}^{n}(ijk)+\theta_{jk}^{i,n}u_{i,n}+\theta_{ki}^{j,n}u_{j,n}+\theta_{ij}^{k,n}u_{k,n}$
$\displaystyle+(\frac{\pi}{2}-\alpha_{ij}^{k,n})\lambda_{ij}+(\frac{\pi}{2}-\alpha_{ki}^{j,n})\lambda_{ki}+(\frac{\pi}{2}-\alpha_{jk}^{i,n})\lambda_{jk}.$

Combining Lemma 3.7 and Lemma 3.10 gives that $\lim\theta^{i,n}_{jk}=0$, the
sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants and the
sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$,
$(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and
$(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge. Combining the continuity of
Milnor’s Lobachevsky function defined by (20) and the definition of the
truncated volume $\mathrm{Vol}(ijk)$ defined by (19), we have that the
sequence $(\mathrm{Vol}^{n}(ijk))_{n\in\mathbb{N}}$ converges. Note that
$\lambda_{ij}=\mathrm{arccosh}I_{ij}$ keeps invariant. Hence,

$\lim F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=\lim\theta_{jk}^{i,n}u_{i,n}+c_{11}$

for some constant $c_{11}$. By (30), we have $u_{i,n}=\log r_{i,n}-\log
r_{i}$. Then

$\displaystyle\lim\theta_{jk}^{i,n}u_{i,n}=$
$\displaystyle\lim\sin\theta^{i,n}_{jk}(\log r_{i,n}-\log r_{i})$
$\displaystyle=$
$\displaystyle\lim\frac{2A_{ijk}^{n}}{l_{ij}^{n}l_{ki}^{n}}\log r_{i,n}$
$\displaystyle=$ $\displaystyle 2c_{4}\lim\frac{\log r_{i,n}}{r_{i,n}}$
$\displaystyle=$ $\displaystyle 0,$

where the equalities (33) and (34) is used in the second line and
$\lim_{x\rightarrow+\infty}\frac{1}{x}\log x=0$ is used in the third line.
Therefore, $\lim F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=c_{11}$. Q.E.D.

The following lemma gives an asymptotic expression of the function
$\mathcal{E}$.

######  Lemma 3.12.

There exists a convergent sequence $\\{D_{n}\\}_{n\in\mathbb{N}}$ such that
the function $\mathcal{E}$ satisfies

$\mathcal{E}(u_{n})=D_{n}+2\pi\left(u_{i^{*},n}\chi(S)+\sum_{j\in
V}(u_{j,n}-u_{i^{*},n})\right).$

###### Proof.

By (26), we have

$\displaystyle\mathcal{E}(u_{n})=$ $\displaystyle-\sum_{\\{ijk\\}\in
F}F_{ijk}(u_{i,n},u_{j,n},u_{k,n})+2\pi\sum_{j\in V}u_{j,n}$ $\displaystyle=$
$\displaystyle-\sum_{\\{ijk\\}\in
F}F_{ijk}((u_{i,n},u_{j,n},u_{k,n})-u_{i^{*},n}(1,1,1))-\pi|F|u_{i^{*},n}+2\pi\sum_{j\in
V}u_{j,n}$ $\displaystyle=$ $\displaystyle
D_{n}-\pi(2|V|-2\chi(S))u_{i^{*},n}+2\pi\sum_{j\in V}u_{j,n}$ $\displaystyle=$
$\displaystyle D_{n}+2\pi\left(u_{i^{*},n}\chi(S)+\sum_{j\in
V}(u_{j,n}-u_{i^{*},n})\right),$

where $D_{n}=-\sum_{\\{ijk\\}\in
F}F_{ijk}((u_{i,n},u_{j,n},u_{k,n})-u_{i^{*},n}(1,1,1))$, the equation (22) is
used in the second line and $2|V|-|F|=2\chi(S)$ is used in the third line. The
sequence $\\{D_{n}\\}_{n\in\mathbb{N}}$ converges by Corollary 3.6 and
Corollary 3.11. Q.E.D.

The following lemma gives the influence of the sequence
$(u_{n})_{n\in\mathbb{N}}$ on the area $A_{ijk}$ of a decorated triangle
$\\{ijk\\}$.

######  Lemma 3.13.

For a discrete conformal factor $u\in\mathbb{R}^{V}$, let $\mathcal{T}$ be a
weighted Delaunay triangulation of the decorated PE surface
$(S,V,dist_{S}(u),r(u))$. Assume the sequences
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$,
$(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge for $\\{ijk\\}$ in
$\mathcal{T}$ with edge lengths $l_{ij}^{n},l_{jk}^{n},l_{ki}^{n}$.

(a):

If $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges, there exists a
convergent sequence of real numbers $(C_{n})_{n\in\mathbb{N}}$ such that

(37) $\log A_{ijk}^{n}=C_{n}+2u_{i^{*},n}.$

(b):

If $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges to $+\infty$, there
exists a convergent sequence of real numbers $(C_{n})_{n\in\mathbb{N}}$ such
that

(38) $\log A_{ijk}^{n}=C_{n}+u_{i,n}+u_{i^{*},n}.$

###### Proof.

Applying (31) to $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$ gives

$\displaystyle A^{n}_{ijk}=$
$\displaystyle\frac{1}{2}l_{ij}^{n}l_{jk}^{n}\sin\theta_{ki}^{j,n}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sin\theta_{ki}^{j,n}\sqrt{r^{2}_{i}e^{2u_{i,n}}+r^{2}_{j}e^{2u_{j,n}}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{(u_{i,n}+u_{j,n})}}$
$\displaystyle\times\sqrt{r^{2}_{j}e^{2u_{j,n}}+r^{2}_{k}e^{2u_{k,n}}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}+u_{k,n})}}.$

Then

$\displaystyle\log A^{n}_{ijk}=$
$\displaystyle\log(\frac{1}{2}\sin\theta_{ki}^{j,n})+2u_{i^{*},n}$
$\displaystyle+\frac{1}{2}\log(r^{2}_{i}e^{2(u_{i,n}-u_{i^{*},n})}+r^{2}_{j}e^{2(u_{j,n}-u_{i^{*},n})}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{(u_{i,n}-u_{i^{*},n})+(u_{j,n}-u_{i^{*},n})})$
$\displaystyle+\frac{1}{2}\log(r^{2}_{j}e^{2(u_{j,n}-u_{i^{*},n})}+r^{2}_{k}e^{2(u_{k,n}-u_{i^{*},n})}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}-u_{i^{*},n})+(u_{k,n}-u_{i^{*},n})})$
$\displaystyle=$
$\displaystyle\log(\frac{1}{2}\sin\theta_{ki}^{j,n})+u_{i,n}+u_{i^{*},n}$
$\displaystyle+\frac{1}{2}\log(r^{2}_{i}+r^{2}_{j}e^{2(u_{j,n}-u_{i^{*},n})-2(u_{i,n}-u_{i^{*},n})}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{-(u_{i,n}-u_{i^{*},n})+(u_{j,n}-u_{i^{*},n})})$
$\displaystyle+\frac{1}{2}\log(r^{2}_{j}e^{2(u_{j,n}-u_{i^{*},n})}+r^{2}_{k}e^{2(u_{k,n}-u_{i^{*},n})}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}-u_{i^{*},n})+(u_{k,n}-u_{i^{*},n})}).$

If the sequence $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges, then $\log
A_{ijk}^{n}=C_{n}+2u_{i^{*},n}$. If the sequence
$(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges to $+\infty$, then the
sequence $(\theta_{ki}^{j,n})_{n\in\mathbb{N}}$ converges to a non-zero
constant in $(0,\pi)$ by Lemma 3.10. This implies $\log
A_{ijk}^{n}=C_{n}+u_{i,n}+u_{i^{*},n}$. In both cases, the sequence
$(C_{n})_{n\in\mathbb{N}}$ converges. Q.E.D.

###  3.4. Proof of Theorem 3.4

Let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be an unbounded “good” sequence. Suppose
$\chi(S)<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{A}$. Combining $\chi(S)<0$ and Lemma 3.12, we just need to prove
that $\lim_{n\rightarrow+\infty}u_{i^{*},n}=-\infty$. By the definition of
“good” sequence, the sequence $\left(\sum_{j\in
V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ converges to a finite number
or diverges properly to $+\infty$.

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$
converges to a finite number, then the sequence
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges for all $j\in V$. Since the
sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ lies in $\mathcal{A}$, the area
$A_{ijk}$ of each triangle is bounded from above. This implies
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ is bounded from above by (37). Then
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ converges to a finite number or diverges
properly to $-\infty$. Suppose $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ converges
to a finite number. Since $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges
for all $j\in V$, then $\\{u_{j,n}\\}_{n\in\mathbb{N}}$ are bounded for all
$j\in V$, which implies $\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded. This
contradicts the assumption that $\\{u_{n}\\}_{n\in\mathbb{N}}$ is unbounded.
Therefore, the sequence $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly
to $-\infty$.

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ diverges
properly to $+\infty$, then there exists at least one vertex $i\in V$ such
that the sequence $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges properly
to $+\infty$. By Corollary 3.6, the sequences
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ and
$(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge for $j\sim i$ and $k\sim i$.
Since the area $A_{ijk}$ of each triangle is bounded from above, thus
$u_{i,n}+u_{i^{*},n}\leq C$ and $u_{j,n}+u_{i^{*},n}\leq C$ by (38), where $C$
is a constant. Then $(u_{i,n}-u_{i^{*},n})+2u_{i^{*},n}\leq C$. This implies
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly to $-\infty$. Q.E.D.

######  Remark 3.14.

For the case $\chi(S)>0$, Kouřimská [17, 16] gave the existence of PE metrics
with constant discrete Gaussian curvatures. However, we can not get similar
results. The main difference is that the edge length defined by (5) involves
the square term of discrete conformal factors, such as $e^{2u_{i}}$, while the
edge length defined by the vertex scalings only involves the mixed product of
the first order terms, i.e., $e^{u_{i}+u_{j}}$. Indeed, in this case, we can
define the set $\mathcal{A}_{+}=\\{u\in\mathbb{R}^{V}|A_{tot}(u)\geq 1\\}$,
which is an unbounded closed subset of $\mathbb{R}^{V}$. Under the conditions
that $\chi(S)>0$ and the function $\mathcal{E}(u)$ attains a minimum in the
set $\mathcal{A}_{+}$, Lemma 3.2 still holds. Using Theorem 3.3, we just need
to prove Theorem 3.4 under the condition $\chi(S)>0$. However, we can not get
a good asymptotic expression of the area $A_{ijk}$. The asymptotic expression
of the area $A_{ijk}$ in (38) involves $u_{i,n}+u_{i^{*},n}$, which is not
enough for this case.

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# The variational theorem for the scattering length in low dimensions and its
applications to universal systems

Alexander Yu. Cherny<EMAIL_ADDRESS>Bogoliubov Laboratory of
Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna,
Moscow region, Russia

###### Abstract

The variational theorem for the scattering length [1] is extended to one and
two dimensions. It is shown that the arising singularities can be treated in
terms of generalized functions. The variational theorem is applied to a
universal many-body system of spinless bosons. The extended Tan adiabatic
sweep theorem is obtained for interacting potentials of arbitrary shape with
the variation of the one-particle dispersion. The pair distribution function
is calculated at short distances by means of the variation of the potential.
The suggested scheme is based on simple quantum mechanics; it is physically
transparent and free from any divergence.

##  I Introduction

For cold atoms, interactions between particles are governed by the low-
momentum $s$-wave scattering amplitude, which is determined by the scattering
length [2, 3, 4]. This property is called universality. When the scattering
length becomes very large, many-body systems are close to the unitary regime,
where the mean-field theoretical description becomes irrelevant and other
approaches are needed [5, 6]. A phenomenological thermodynamic approach is to
consider the scattering length, which can be controlled with the Feshbach
resonance, as a thermodynamic property. The Tan adiabatic sweep theorem [7, 8]
relates the increment of energy to infinitesimal change of the scattering
length when it changes slowly.

In the previous publication [9], we derived the Tan theorem for a system of
spinless bosons in three dimension (3D) by means of the variational theorem
for the scattering length [1]. In this paper, we generalize the variational
theorem to lower dimensions and apply it to the Tan theorem in 1D and 2D.
Besides, we calculate the short-range part of the pair distribution function
in these dimensions.

The 2D variational theorem with respect to interparticle interaction of
arbitrary shape was considered in the paper [10]. Here we suggest the full
form of the theorem in the low dimensions, which includes the variations of
both one-particle dispersion and interaction. The theorem involves the Fourier
transform of the scattering part of two-body wave function, which does not
tend to zero at large distances in 1D nd 2D. Below we show how to overcome
this difficulty with the help of generalized functions.

The standard approach to universal systems is to use the pseudopotential,
proportional to the $\delta$-function. This scheme is quite convenient
technically but leads to divergencies. Special methods are needed to get rid
of the divergencies [11, 12, 13]. In the previous papers [1, 10, 14], we
developed the approach, which is free from divergencies and applicable to
short-range interactions of arbitrary shape. Conceptually, the method of this
paper is quite similar: we apply the variational theorem for an arbitrary
potential and generalize this way the Tan adiabatic sweep theorem for one [15]
and two [16] dimensions, see Eq. (24). Equation (26) for the pair distribution
function generalizes our previous results [1] to one and two dimensions.

The paper is organize as follows. In the next section, the variational theorem
in arbitrary dimension is proved. In Sec. III, the generalized Lippmann-
Schwinger equation in 2D is written down explicitly. In order to verify the
variational theorem with a specific example, we consider an exactly solvable
case of the circle $\delta$-potential well. We apply the variational theorem
to find the long-range momentum asymptotics of the average occupation numbers
and short-range behaviour of the pair distribution function in Sec. IV. In
Conclusion, the main results are discussed.

##  II The variational theorem for the scattering length in arbitrary
dimension

Let us consider two particles of equal masses, interacting with the potential
${V}(r)$. It is assumed to be of the short-range type, that is, it decreases
at infinity as $1/r^{\alpha}$ with the exponent $\alpha>3$ in 1D and 3D and
$\alpha>2$ in 2D or faster. The dispersion relation is assumed to be of
general form $T(p)$, which means that the operator of kinetic energy in the
momentum representation is equal to $T(p)$. It is supposed to be a function of
the absolute value of $\bm{p}$ and quite close to the usual free-particle
dispersion at small momenta:

$\displaystyle T(p)-\frac{p^{2}}{2m}=O(p^{4}).$ (1)

Here the symbol $O(z)$ denotes the terms of order $z$ or smaller. We also
impose the condition $T(p)-\frac{p^{2}}{2m}=O(1/p^{2})$ for large momenta.

The Hamiltonian of two particles takes the form
$\hat{H}=T(p_{1})+T(p_{2})+{V}(|\bm{r}_{1}-\bm{r}_{2}|)$. As usual, it is
convenient to introduce two pairs of the canonically conjugate variables: the
relative momentum $\bm{p}=(\bm{p}_{1}-\bm{p}_{2})/2$ and position
$\bm{r}=\bm{r}_{1}-\bm{r}_{2}$, and the total momentum
$\bm{Q}=\bm{p}_{1}+\bm{p}_{2}$ and the center-of-mass position
$\bm{R}=(\bm{r}_{1}+\bm{r}_{2})/2$. Then the kinetic energy of the two
particles reads
$T\left(\frac{\bm{Q}}{2}+\bm{p}\right)+T\left(\frac{\bm{Q}}{2}-\bm{p}\right)$.
Although the center-of-mass motion no longer separates in general, the total
momentum $\bm{Q}=\bm{p}_{1}+\bm{p}_{2}$ is still conserved, and thus it is a
good quantum number. This allows us to look for eigenfunctions in the form
$\varPhi(\bm{r}_{1},\bm{r}_{2})=\exp(i\bm{Q}\cdot\bm{R}/\hbar)\varphi_{\bm{Q}}(\bm{r})$
with $\varphi_{\bm{Q}}$ being the wave function of the relative motion. In
general, $\varphi_{\bm{Q}}(\bm{r})$ depends on the total momentum, while for
the usual dispersion $p^{2}/(2m)$ it is independent of $\bm{Q}$.

We consider the two-particle scattering problem with zero total energy and
momentum. Then kinetic energy is equal to $2T(p)$, and the two-body
Schrödinger equation takes the form

$2T\left(-i\hbar\nabla\right)\varphi(r)+{V}(r)\varphi(r)=0.$ (2)

For a short-range potential and free-particle dispersion obeying the
restriction (1), the radially symmetric solution satisfies the boundary
conditions for $r\to\infty$

$\varphi(r)\simeq\begin{cases}1-a/r,&D=3,\\\ 1-\ln r/\ln a,&D=2,\\\
1-r/a,&D=1.\end{cases}$ (3)

The parameter $a$ is called the scattering length. It can take arbitrary real
value in 1D and 3D, while it is always positive in 2D [10]. In two dimensions,
we follow the definition given in Refs. [17, 10, 14, 16]. At zero radius, the
boundary condition is $|\varphi(r=0)|<\infty$ in 2D and 3D, while in 1D one
can put [18] $\left.\frac{\partial\varphi(r)}{\partial r}\right|_{r=0}=0$ when
the potential is not very singular at $r=0$: $|V(r)|<C/r^{s}$ with $s<1$.

It is convenient to separate the scattering part $\psi(r)$ of the wave
function defined as

$\displaystyle\varphi(r)=1+\psi(r).$ (4)

Multiplying Eq. (2) by $\varphi(r)$, using the identity
$T\left(-i\hbar\nabla\right)1=0$, and integrating over the whole real space
yield

${U}(0)=\int{d}^{D}r\left[\psi(r)2T\left(-i\hbar\nabla\right)\psi(r)+{V}(r)[\varphi(r)]^{2}\right],$
(5)

where $U(0)$ is the zero component of the scattering amplitude

$\displaystyle U(p)=\int{d}^{D}r\,{V}(r)\varphi(r)e^{-i\bm{p}\cdot\bm{r}}.$
(6)

Equation (5) tells us that the zero-momentum scattering amplitude is a
functional of the wave function and its stationary solution obeying the
condition $\delta U(0)=0$ satisfies the Schrödinger equation (2). It follows
that when we vary $U(0)$ with respect to a parameter of the Hamiltonian at the
stationary solution, we should not take into account a variation of the wave
function. Then we arrive at the variational theorem [1] in arbitrary
dimensions

$\displaystyle\delta{U}(0)=\int{d}^{D}r\Bigl{[}\psi(r)\delta
T\left(-i\hbar\nabla\right)\psi(r)+\delta{V}(r)\varphi^{2}(r)\Bigr{]}.$ (7)

It is convenient to write the first term in the Fourier representation

$\displaystyle\delta{U}(0)=\int\frac{{d}^{D}p}{(2\pi)^{D}}\,2\delta
T({p})\psi^{2}(p)+\int{d}^{D}r\,\delta{V}(r)\varphi^{2}(r),$ (8)

where the Fourier transform of the scattering part of the wave function is
given by

$\displaystyle\psi_{p}=-\frac{U(p)}{2T({p})}$ (9)

Equation (9) is nothing else but the Schrödinger equation (2) in the Fourier
representation.

Some caveats should be given here. In one and two dimensions, $\psi(r)$ does
not tend to zero when $r\to\infty$ in accordance with Eq. (3). This
corresponds to divergence of the Fourier transform of Eq. (9) at zero momentum
when $U(0)\not=0$. For this reason, its Fourier transform should be given in
terms of generalized functions, and the appropriate regularization should be
done [19]. The singular part of $\psi_{p}$ is proportional to $1/p^{2}$, and
the Fourier transform of $1/p^{2}$ can be understood as the solution of
equation $\nabla^{2}f(r)=-\delta(\bm{r})$ with $\delta(\bm{r})$ being the
Dirac $\delta$ function in $D$ dimension. Then the function $f(r)$ is a sum of
a particular solution and arbitrary solutions of the homogeneous equation
$\nabla^{2}f(r)=0$, and we get an indeterminacy due to the regularization.
However, the indeterminacy can be eliminated by the boundary condition (3). We
obtain $f(r)=-r/2$, $f(r)=-\ln r/(2\pi)$, and $f(r)=1/(4\pi r)$ in one, two,
and three dimensions, respectively. This reasoning is analogous to that of
used for constructing Green functions in the Lippmann-Schwinger equation.

It follows from the above considerations that the main singularity
$-U(0)m/(\hbar^{2}p^{2})$ at $p=0$ in the r.h.s. of Eq. (9) determines the
main asymptotics of $\psi(r)$ for $r\to\infty$. Using the definition of the
scattering length (3), we arrive at the relation between the scattering length
and the scattering amplitude at zero momentum:

$\displaystyle U(0)=$ $\displaystyle\frac{\hbar^{2}}{m}\times\begin{cases}4\pi
a,&D=3,\\\ -2\pi/\ln a,&D=2,\\\ -2/a,&D=1.\end{cases}$ (10)

The variation of the scattering amplitude with respect to the interaction
potential in two and three dimensions was considered in Refs. [10] and [20],
respectively. The variation with the both one-particle dispersion and
interaction potential in three dimension was found in Ref. [1].

Finally, the variational theorem (8) in arbitrary dimension reads

$\displaystyle\frac{\delta U(0)}{\delta T({p})}=$
$\displaystyle\frac{1}{(2\pi)^{D}}\frac{U^{2}(p)}{2T^{2}({p})}=\frac{m^{2}U^{2}(p)}{2^{D-1}\pi^{D}\hbar^{4}p^{4}},$
(11) $\displaystyle\frac{\delta U(0)}{\delta V(r)}=$
$\displaystyle\varphi^{2}(r).$ (12)

The second equality in Eq. (11) assumes that the variation is taken in the
vicinity of the usual free-particle dispersion.

The short-range potential $V(r)$ is supposed to be localized within its
characteristic radius $r_{0}$. Then the asymptotics (3) is realized for $r\gg
r_{0}$, and, besides, $U(p)\simeq U(0)$ when $p\ll 1/r_{0}$.

##  III The generalized Lippmann-Schwinger equation for arbitrary one-particle
dispersion in two dimensions

The considerations of Sec. II allow us to correctly define the corresponding
form of the Lippmann-Schwinger equation for arbitrary one-particle dispersion.
Equation (9) implies that $\psi(r)$ is the convolution of $V(r)\varphi(r)$ and
the Green function, which is the Fourier transform of $1/T(p)$ with arbitrary
$T(p)$. In this section, we consider as an example the generalized Lippmann-
Schwinger equation in two dimensions and its exact analytical solution for a
specific interaction potential.

Separating the singular part $2m/(\hbar^{2}p^{2})$ in $\psi_{p}$ (9), taking
the Fourier transformation, and integrating over the polar angle, we obtain
with the help of Eqs. (4) and (6)

$\displaystyle\varphi(r)=$ $\displaystyle
1+\frac{m}{\hbar^{2}}\int_{0}^{\infty}dy\,yV(y)\varphi(y)\Big{[}\theta(r-y)\ln
r$ $\displaystyle+\theta(y-r)\ln y+I(r,y)\Big{]},$ (13) $\displaystyle
I(r,y)=$
$\displaystyle\frac{\hbar^{2}}{2m}\int_{0}^{\infty}dp\,p\left(\frac{2m}{\hbar^{2}p^{2}}-\frac{1}{T(p)}\right)J_{0}(pr)J_{0}(py),$
(14)

where $\theta(z)$ and $J_{0}(z)$ are the Heaviside step function and Bessel
function of zero order, respectively. Due to the condition (1), the integrand
in Eq. (14) is regular at zero momentum.

The generalized Lippmann-Schwinger equation (13) admits an analytical solution
for the interaction potential

$\displaystyle V(r)=\frac{\hbar^{2}}{mr_{0}\varkappa}\delta(r-r_{0}).$ (15)

Here the dimensionless control parameter $\varkappa$ determines the strength
of interaction; it can be positive or negative. The presence of the $\delta$
function enables us to get rid of the integral, and the integral equation (13)
becomes algebraic. Substituting Eq. (15) into the generalized Lippmann-
Schwinger equation, we obtain the solution for $\varphi(r_{0})$

$\displaystyle\varphi(r_{0})=\frac{\varkappa}{\varkappa-\ln
r_{0}-I(r_{0},r_{0})}.$ (16)

Once $\varphi(r_{0})$ is known, we find the scattering amplitude for arbitrary
one-particle dispersion with the definition (6)

$\displaystyle U(p)=$ $\displaystyle U(0)J_{0}(pr_{0}),$ (17) $\displaystyle
U(0)=$ $\displaystyle\frac{2\pi\hbar^{2}}{m\varkappa}\varphi(r_{0}).$ (18)

Now the scattering amplitude of the standard Schrödinger equation (2) is
obtained from Eqs. (16), (17), and (18) in the particular case
$T(p)=\hbar^{2}p^{2}/(2m)$, which corresponds to $I(r_{0},r_{0})=0$. With the
help of Eq. (10), we get the scattering length

$\displaystyle a=r_{0}e^{-\varkappa}.$ (19)

It is not difficult to find explicitly the variation $\delta U(0)$ with
respect to $\delta T(p)$ and directly check the variation theorem (11) in the
particular case of interaction (15), see Appendix A.

The behaviour of the scattering amplitude for the potential (15) is shown in
Fig. 1a. The ratio $U(p)/U(0)$ is the Bessel function of zero order in
accordance with Eq. (17). We also calculate numerically the scattering
amplitude (6) for the Gaussian potential

$\displaystyle
V(r)=\frac{\hbar^{2}\alpha}{mr_{0}^{2}}\exp\left(-\frac{r^{2}}{r_{0}^{2}}\right),$
(20)

see the details in Ref. [18]. Here $\alpha$ is the dimensionless control
parameter. We choose $\alpha=-9$, for which $a/r_{0}=0.3289\ldots$. The
results are represented in Fig. 1b. The approximation $U(p)\simeq U(0)$ is
satisfied only when $pr_{0}\ll 1$, and at large wave vectors $p\gg 1/r_{0}$,
$U(p)$ tends to zero, which is a typical behaviour for short-range potentials.

Figure 1:  The scattering amplitude (6) (in units of
$U(0)={-2\pi\hbar^{2}}/{m\ln a}$) as a function of the wavenumber (in units of
the inverse potential radius $1/r_{0}$) for (a) the circle $\delta$-potential
well (15) and (b) the Gaussian potential (20) with the control parameter
$\alpha=-9$. The approximation $U(p)\simeq U(0)$ is satisfied only when $p\ll
1/r_{0}$, and the scattering amplitude $U(p)$ always falls off to zero at
large wave vectors $p\gg 1/r_{0}$.

##  IV The Tan adiabatic sweep theorem and the pair distribution function in
one and two dimensions

The Tan adiabatic sweep theorem [7, 8] relates the derivative of a
thermodynamic potential of a universal system with respect to the scattering
length to the asymptotic behaviour of the average occupation numbers for large
momenta. In our previous paper [9], a simple derivation of the Tan theorem is
given for three-dimensional universal systems. Here we consider the ground
state energy of a homogeneous system of interacting spinless bosons in one and
two dimensions.

In general, the ground state energy per $D$-volume can be written down through
the average occupation numbers $n_{p}$ and pair distribution function $g(r)$

$\displaystyle\varepsilon=\frac{E}{L^{D}}=\frac{1}{(2\pi)^{D}}\int{d}^{D}p\,T(p)n_{p}+\frac{n^{2}}{2}\int{d}^{D}r\,g(r)V(r)$
(21)

with $L$ and $n=N/L^{D}$ being the system size and density of particles,
respectively. We emphasize that the physical dimensions of the energy per
$D$-volume are different in various dimensions; the same relates to the
density of particles.

In general, both $n_{p}$ and $g(r)$ depend on the one-particle dispersion and
interaction. However, in agreement with the Hellmann-Feynman theorem, one can
take into consideration only the explicit dependence on the parameters while
varying the energy (see the detailed discussion in Ref. [9]):

$\displaystyle\frac{\delta\varepsilon}{\delta T(p)}=$
$\displaystyle\,\frac{n_{p}}{(2\pi)^{D}},$ (22)
$\displaystyle\frac{\delta\varepsilon}{\delta V(r)}=$
$\displaystyle\,\frac{n^{2}}{2}g(r).$ (23)

###  IV.1 The long-range asymptotics of the occupation numbers

We find the momentum distribution at high momenta. The scattering amplitude
$U(p)$ is supposed to be found from the ordinary Schrödinger equation (2) with
the usual dispersion relation.

Technically, it is convenient to hold the relation (10) between $U(0)$ and $a$
while varying the dispersion relation. Then we vary the one-particle
dispersion $T(p)$ near $T(p)=\hbar^{2}p^{2}/(2m)$ but keep the mass constant.
For universal systems, the variation of the ground-state energy depends on
$\delta T(p)$ only through the variation of the scattering length:
$\frac{\delta\varepsilon}{\delta T(p)}=\frac{\partial\varepsilon}{\partial
a}\frac{\delta a}{\delta T(p)}$. Equating this expression and Eq. (22), and
using Eq. (10) and the variation theorem (11), we arrive at the relations

$\displaystyle
p^{4}n_{p}=\frac{4m}{\hbar^{2}}\frac{\partial\varepsilon}{\partial
a}\frac{U^{2}(p)}{U^{2}(0)}\times\begin{cases}\pi a,&D=2,\\\
1,&D=1.\end{cases}$ (24)

where $U(p)$ is the scattering amplitude (6) for the standard Schödinger
equation (2) with the usual dispersion
$T(-i\hbar\nabla)=-\hbar^{2}\nabla^{2}/(2m)$.

Thus we obtain the generalization of Tan’s adiabatic sweep theorem for
arbitrary short-range potentials in 1D and 2D. Equation (24) is valid for
arbitrary $p\gg 1/\xi$, where $\xi$ is a parameter with the dimension of
length, which determines the characteristic scale of the many-body effects.
Within the mean-field approximation, $\xi=\hbar/\sqrt{\mu m}$ is the healing
length with $\mu$ being the chemical potential, and the approximation is
applicable when $\xi\gg n^{1/D}$. In the dilute 2D Bose gas, when the gas
parameter $na^{2}$ is small, we have [21, 10, 22]
$\xi\simeq\frac{1}{2}\sqrt{\frac{\ln na^{2}}{-\pi n}}$. In one dimension, the
healing length is given by [23] $\xi\simeq\sqrt{\frac{-a}{2n}}$, and the mean-
field approach works when the density is sufficiently big: $\sqrt{-na}\gg 1$.
For large $na^{2}$ in the 2D case, the ground state becomes unstable against
cluster formation and further crystallization [24, 25]. For small $-na$ in 1D
case, the parameter $\xi$ is of order of mean distance between particles:
$\xi\sim 1/n$. Within the range $1/\xi\ll p\ll 1/r_{0}$, the scattering
amplitude $U(p)$ is almost constant: $U(p)\simeq U(0)$, see the discussion in
Sec. II above. Then we arrive at Tan’s theorem [7]

$\displaystyle p^{4}n_{p}\simeq{\cal
C}=\frac{4m}{\hbar^{2}}\frac{\partial\varepsilon}{\partial
a}\times\begin{cases}\pi a,&D=2,\\\ 1,&D=1,\end{cases}$ (25)

where ${\cal C}$ is the parameter introduced by Tan [8] and called Tan’s
contact. Equation (25) can formally be derived within the pseudopotential
approach, where the relation $U(p)=U(0)$ is exact. For a “usual” potential of
arbitrary shape, the limit $p^{4}n_{p}$ is equal to zero when $p\to\infty$,
because $U(p)$ tends to zero at large momenta, see Fig. 1.

###  IV.2 The short-range spatial correlations

The short-range spatial correlations for a universal system are obtained in
the same manner [1, 26, 9]. We find from Eq. (23) that
$\frac{\delta\varepsilon}{\delta
V(r)}=\frac{n^{2}}{2}g(r)=\frac{\partial\varepsilon}{\partial a}\frac{\delta
a}{\delta V(r)}$. Using the relation (10) and variational theorem (12) finally
yields

$\displaystyle
g(r)=\frac{m}{\hbar^{2}n^{2}}\frac{\partial\varepsilon}{\partial
a}\varphi^{2}(r)\times\begin{cases}a(\ln a)^{2}/\pi,&D=2,\\\
a^{2},&D=1.\end{cases}$ (26)

As one can see, at short distances the pair distribution function is
proportional [1, 10] to the squared wave function of the standard Schrodinger
equation (2). We note that in two dimension, the renormalized wave function
$-\varphi(r)\ln a$ obeys to the boundary condition $-\varphi(r)\ln
a\simeq\ln(r/a)$ in accordance with Eq. (3). The prefactor can be written in
terms of Tan’s contact (25)

$\displaystyle g(r)=\frac{{\cal
C}}{4n^{2}}\varphi^{2}(r)\times\begin{cases}(\ln a)^{2}/\pi^{2},&D=2,\\\
a^{2},&D=1.\end{cases}$ (27)

Equations (26) and (27) are valid at short distances $r\ll\xi$. It follows
from the asymptotics (3) that in two dimensions, $\ln^{2}a\,\varphi^{2}(r)$ is
approximately equal to $\ln^{2}(r/a)$ for $r_{0}\ll r\ll\xi$. In one
dimension, $a^{2}\varphi^{2}(r)\simeq(a-r)^{2}$ under the same conditions.
Then the relation (27) leads to

$\displaystyle g(r)=\frac{{\cal
C}}{4n^{2}}\times\begin{cases}\ln^{2}(r/a)/\pi^{2},&D=2,\\\
(a-r)^{2},&D=1.\end{cases}$ (28)

within the range $r_{0}\ll r\ll\xi$. These results are in consistency with
Refs. [16, 15] obtained in the framework of the pseudopotential approach.

Note that Eqs. (24) and (26) are also valid in the canonical and grand
canonical ensembles if the derivative of the energy is replaced by the
derivative of the corresponding thermodynamic potential, see the discussion in
Ref. [9].

##  V Conclusion

We generalize the full form of the variational theorem for the scattering
length [1] to arbitrary dimension, where the variation is taken with respect
to one-particle dispersion (11) and interaction potential (12), and verify it
in the particular case of the circle $\delta$-potential well (15), see
Appendix A.

With the help of the variational theorem, the Tan adiabatic sweep theorem is
extended to short-range potentials of arbitrary shape in 1D and 2D for a
universal system of spinless bosons, see Eq. (24). This equation gives us the
long-range asymptotics of the mean occupation numbers, which contains the
additional factor proportional to the squared scattering amplitude (6) for the
standard Schrödinger equation (2) with the dispersion
$T(-i\hbar\nabla)=-\hbar^{2}\nabla^{2}/(2m)$. Equation (24) is applicable for
wave vectors more than $1/\xi$, where $\xi$ is the characteristic scale of the
many-body effects (see the discussion in Sec. IV.1). By contrast, the Tan
relation (25) is valid for $1/\xi\ll p\ll 1/r_{0}$, where $r_{0}$ is the
radius of the interaction potential. For the non-polarized two-component
fermions, the analogous equations can also be written down with the same
method.

The short-range behaviour of the pair distribution function (26) is obtained
by means of the variation with respect to interaction potential (12). The pair
distribution function is known at all distances $r\ll\xi$, while the
pseudopotential approach leads to Eq. (28), which described $g(r)$ beyond the
radius of interaction potential $r_{0}\ll r\ll\xi$.

As a prospect, one can calculate the mean kinetic and interaction energies for
a homogeneous universal many-body systems in the low dimensions by analogy
with the paper [9].

##  VI Acknowledgement

The author acknowledges support from the JINR–IFIN-HH projects.

##  Appendix A The variational theorem for the circle delta-potential well

In this appendix, we prove straightforwardly the variational theorem (11) in
two dimensions for the potential (15). As in Sec. IV.1, it is technically
convenient to keep the mass constant to hold the relation (10) between $U(0)$
and $a$.

We obtain from Eqs. (16) and (18)

$\displaystyle\frac{\delta U(0)}{\delta
T(p)}=\frac{2\pi\hbar^{2}}{m\varkappa}\frac{\delta\varphi(r_{0})}{\delta
T(p)}=\frac{2\pi\hbar^{2}}{m\varkappa^{2}}\varphi^{2}(r_{0})\frac{\delta
I(r_{0},r_{0})}{\delta T(p)}.$ (29)

It follows from the definition (14) that

$\displaystyle\frac{\delta I(r_{0},r_{0})}{\delta
T(p)}=\frac{\hbar^{2}}{2m}\frac{J_{0}^{2}(pr_{0})}{2\pi T^{2}(p)},$ (30)

because the volume element in two dimensions is given by $d^{2}p=2\pi p\,dp$.
After substituting Eq. (30) into Eq. (29), we are left with

$\displaystyle\frac{\delta U(0)}{\delta
T(p)}=\frac{\hbar^{4}}{m^{2}\varkappa^{2}}\frac{\varphi^{2}(r_{0})J_{0}^{2}(pr_{0})}{2T^{2}(p)},$
(31)

which coincides with the relation (11) for $D=2$ by virtue of Eqs. (17) and
(18).

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