# Prescribing discrete Gaussian curvature on polyhedral surfaces

Xu Xu, Chao Zheng

###### Abstract

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo [26]
is a straightforward discretization of smooth conformal structures on
surfaces. Combinatorial $\alpha$-curvature for vertex scaling of piecewise
linear metrics on surfaces is a discretization of Gaussian curvature on
surfaces. In this paper, we investigate the prescribing combinatorial
$\alpha$-curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu’s
discrete conformal theory [19] for piecewise linear metrics on surfaces and
variational principles with constraints, we prove some Kazdan-Warner type
theorems for prescribing combinatorial $\alpha$-curvature problem, which
generalize the results obtained in [19, 40] on prescribing combinatorial
curvatures on surfaces. Gu-Luo-Sun-Wu [19] conjectured that one can prove
Kazdan-Warner’s theorems in [24, 25] via approximating smooth surfaces by
polyhedral surfaces. This paper takes the first step in this direction.

MSC (2020): 52C26

Keywords: Polyhedral metrics; Vertex scaling; Combinatorial curvature;
Variational principle

##  1 Introduction

###  1.1 Motivation

A classical problem in differential geometry is prescribing Gaussian curvature
on closed surfaces, which asks the follow question.

Prescribing Gaussian curvature problem. Let $S$ be a smooth connected closed
surface. What functions could be realized as Gaussian curvatures of Riemannian
metrics on $S$?

This problem has been extensively studied by lots of mathematicians by
deforming Riemannian metric in its conformal structure. See, for instance, [3,
6, 30, 24, 25] and others.

Polyhedral metric on surfaces is a discrete analogue of the Riemannian metric
on surfaces, which is a constant curvature ($1$, $0$ or $-1$) metric with cone
singularities. The classical discrete Gaussian curvature for a polyhedral
metric on a surface is the corresponding discrete analogue of the smooth
Gaussian curvature, which is defined to be the angle defect at the cone
points. Prescribing discrete Gaussian curvature problem on polyhedral surface
asks the following question parallelling to the classical prescribing Gaussian
curvature problem on smooth surfaces.

Prescribing discerete Gaussian curvature problem. Suppose $S$ is a connected
closed surface and $V$ is a finite nonempty subset of $S$. What functions
defined on $V$ could be realized as discrete Gaussian curvatures of polyhedral
metrics on $(S,V)$?

In discrete conformal geometry, parallelling to the smooth case, the
prescribing discrete Gaussian curvature problem on surfaces is usually studied
by deforming a polyhedral metric in its discrete conformal structure, which is
a discrete analogue of the smooth conformal structure on surfaces defining
polyhedral metrics by functions defined on the cone points. The discrete
conformal structures on surfaces that have been extensively studied in the
history include Thurston’s circle packings [36], Luo’s vertex scaling [26] and
others. For Thurston’s circle packings on surfaces, the solution of
prescribing discrete Gaussian curvature problem gives rise to the famous
Koebe-Andreev-Thurston Theorem, which plays a fundamental role in the study of
geometry and topology of 3-dimensional manifolds [36]. For vertex scaling of
piecewise linear and piecewise hyperbolic metrics on surfaces, the recent
perfect solution of prescribing discrete Gaussian curvature problem by Gu-Luo-
Sun-Wu [19] and Gu-Guo-Luo-Sun-Wu [18] provides a new constructive proof of
the classical uniformization theorem on closed surfaces with genus $g\geq 1$,
which is computable and has lots of applications [10, 21, 28, 45, 20, 35]. In
the framework of vertex scaling, Springborn [34] recently proved that the
discrete uniformization theorem on the sphere is equivalent to Rivin’s
realization theorem for ideal hyperbolic polyhedra [33].

However, the classical discrete Gaussian curvature is not a proper
discretization of the smooth Gaussian curvature on surfaces. For example, it
is scaling invariant, which is different from the transformation of smooth
Gaussian curvatures under scaling of Riemannian metrics, and does not
approximate the smooth Gaussian curvature pointwisely as the triangulations
become finer and finer [17]. In [40], the first author introduced
combinatorial $\alpha$-curvature for vertex scaling of piecewise linear
metrics (PL metrics for short in the following) on surfaces to avoid the
disadvantages of classical discrete Gaussian curvature alluded to above, where
the rigidity and Yamabe problem for combinatorial $\alpha$-curvature were also
studied. In this paper, we study the prescribing combinatorial
$\alpha$-curvature problem for PL metrics on surfaces and prove some Kazdan-
Warner type theorems. It is conjectured by Gu-Luo-Sun-Wu [19] that one can
prove Kazdan-Warner’s theorems in [24, 25] by approximating smooth surfaces by
polyhedral surfaces involving vertex scaling. This paper takes the first step
in this direction.

###  1.2 Statements of main results

Suppose $S$ is a connected closed surface and $V$ is a finite non-empty subset
of $S$, $(S,V)$ is called a marked surface. A PL metric $d$ on the marked
surface $(S,V)$ is a flat cone metric on $S$ with conic singularities
contained in $V$. A triangulation $\mathcal{T}$ of the marked surface $(S,V)$
is a triangulation of $S$ with the vertex set equal to $V$. Denote a vertex,
an edge and a face in the triangulation $\mathcal{T}$ by
$i,\\{ij\\},\\{ijk\\}$ respectively and denote the sets of edges and faces of
$\mathcal{T}$ by $E_{\mathcal{T}}$ and $F_{\mathcal{T}}$ respectively. A
triangulation $\mathcal{T}$ of $(S,V,d)$ is geometric if every edge in
$E_{\mathcal{T}}$ is a geodesic in the PL metric $d$. A PL metric $d$ on a
triangulated surface $(S,V,\mathcal{T})$ with $\mathcal{T}$ geometric defines
a map $l:E_{\mathcal{T}}\rightarrow(0,+\infty)$ such that
$l_{ij},l_{ik},l_{jk}$ satisfy the triangle inequalities for any triangle
$\\{ijk\\}\in F_{\mathcal{T}}$. The map
$l:E_{\mathcal{T}}\rightarrow(0,+\infty)$ is called as a discrete metric and
the logarithm of the discrete metric $l$, $\lambda_{ij}=2\log l_{ij}$, is
called as the logarithmic length. One can also obtain discrete PL metrics on a
triangulated surface $(S,V,\mathcal{T})$ by gluing triangles in 2-dimensional
Euclidean space isometrically along the edges in pair, which gives rise to PL
metrics on the marked surface $(S,V)$. Note that a PL metric $d$ on a marked
surface $(S,V)$ is intrinsic in the sense that it is independent of the
geometric triangulations of $(S,V,d)$. Every PL metric $d$ on $(S,V)$ has a
Delaunay triangulation $\mathcal{T}$ of $(S,V,d)$ such that each triangle in
$\mathcal{T}$ is Euclidean and the sum of two angles facing each edge is at
most $\pi$. See [2, 5, 19, 32] for further discussion on Delaunay
triangulation of polyhedral surfaces.

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$ and $\mathcal{T}$
is a geometric triangulation of $(S,V,d)$. The classical discrete Gaussian
curvature $K:V\rightarrow(-\infty,2\pi)$ for $(S,V,d)$ is defined as

$K_{i}=2\pi-\sum_{\\{ijk\\}\in F_{\mathcal{T}}}\theta_{i}^{jk}$ (1)

with summation taken over all the triangles at $i\in V$ and $\theta_{i}^{jk}$
being the inner angle of the triangle $\\{ijk\\}\in F_{\mathcal{T}}$ at the
vertex $i$. Sometimes we call $K$ as combinatorial curvature for simplicity.
Note that the classical discrete Gaussian curvature $K$ is intrinsic in the
sense that it is independent of the geometric triangulations of $(S,V,d)$. The
classical combinatorial curvature $K$ for PL metrics on $(S,V)$ satisfies the
following discrete Gauss-Bonnet formula ([7], Proposition 3.1)

$\sum_{i\in V}K_{i}=2\pi\chi(S).$ (2)

The discrete Gauss-Bonnet formula (2) provides a necessary condition for a
function to be the discrete Gaussian curvature of some PL metric on $(S,V)$.
The classical prescribing discrete Gaussian curvature problem on surfaces can
be taken as a converse problem to the discrete Gauss-Bonnet formula (2), which
asks the following question.

######  Question 1.1.

Is the discrete Gauss-Bonnet formula (2) sufficient condition for a function
$K:V\rightarrow(-\infty,2\pi)$ to be the discrete Gaussian curvature of some
PL metric on $(S,V)$?

The prescribing discrete Gaussian curvature problem is usually studied in the
frame of discrete conformality of polyhedral metrics on surfaces. See, for
instance, [36, 19, 18, 26] and others. Vertex scaling of PL metrics on
surfaces introduced by Luo [26] is a straightforward discrete analogue of the
conformal transformation in Riemannian geometry.

######  Definition 1.2 ([26]).

Suppose $l,\widetilde{l}:E_{\mathcal{T}}\rightarrow(0,+\infty)$ are two
discrete PL metrics on a triangulated surface $(S,V,\mathcal{T})$.
$\widetilde{l}$ is called as a vertex scaling of $l$ if there exists a
function $u:V\rightarrow\mathbb{R}$ such that

$\widetilde{l}_{ij}=l_{ij}\exp\left(\frac{u_{i}+u_{j}}{2}\right)$

for any edge $\\{ij\\}\in E_{\mathcal{T}}$.

Among other things, Luo [26] further established a variational principle for
vertex scaling of PL metrics on surfaces and proved that there exists some
combinatorial obstructions for the existence of PL metric with constant
discrete Gaussian curvature $K$ in a discrete conformal class on a
triangulated surface $(S,V,\mathcal{T})$ in the sense of Definition 1.2. This
implies that there exist some combinatorial obstructions for the solvability
of the prescribing discrete Gaussian curvature problem in a discrete conformal
class on a triangulated surface $(S,V,\mathcal{T})$ in the sense of Definition
1.2. To overcome this difficulty, based on Bobenko-Pinkall-Springborn’s
important work [4], Gu-Luo-Sun-Wu [19] introduced the following new definition
of discrete conformality of PL metrics on marked surfaces, which allows the
triangulation of the marked surface to be changed under the Delaunay
condition.

######  Definition 1.3 ([19], Definition 1.1).

Two PL metrics $d,d^{\prime}$ on a marked surface $(S,V)$ are discrete
conformal if there exist a sequence of PL metrics
$d_{1}=d,d_{2},...,d_{m}=d^{\prime}$ on $(S,V)$ and triangulations
$\mathcal{T}_{1},...,\mathcal{T}_{m}$ of $(S,V)$ satisfying

(a)

(Delaunay condition) each $\mathcal{T}_{i}$ is Delaunay in $d_{i}$,

(b)

(Vertex scaling condition) if $\mathcal{T}_{i}=\mathcal{T}_{i+1}$, there
exists a function $u:V\rightarrow\mathbb{R}$, called a conformal factor, so
that if $e$ is an edge in $\mathcal{T}_{i}$ with end points $v$ and
$v^{\prime}$, then the lengths $l_{d_{i}}(e)$ and $l_{d_{i+1}}(e)$ of $e$ in
metrics $d_{i}$ and $d_{i+1}$ are related by

$l_{d_{i+1}}(e)=l_{d_{i}}(e)\exp\left(\frac{u(v)+u(v^{\prime})}{2}\right),$

(c)

if $\mathcal{T}_{i}\neq\mathcal{T}_{i+1}$, then $(S,d_{i})$ is isometric to
$(S,d_{i+1})$ by an isometry homotopic to the identity in $(S,V)$.

Definition 1.3 defines an equivalence relationship for PL metrics on a marked
surface $(S,V)$. The equivalence class of a PL metric $d$ on $(S,V)$ is called
as the discrete conformal class of $d$ and denoted by $\mathcal{D}(d)$. Using
the new discrete conformality in Definition 1.3, Gu-Luo-Sun-Wu [19] perfectly
solved the prescribing discrete Gaussian curvature problem for PL metrics on
closed surfaces in the following well-known theorem, which shows that the
discrete Gauss-Bonnet formula (2) is a necessary and sufficient condition for
a function $\overline{K}:V\rightarrow(-\infty,2\pi)$ to be the classical
discrete Gaussian curvature of some PL metric on $(S,V)$.

######  Theorem 1.4 ([19] Theorem 1.2).

Suppose $(S,V)$ is a closed connected marked surface and $d$ is a PL metric on
$(S,V)$. Then for any $\overline{K}:V\rightarrow(-\infty,2\pi)$ with
$\sum_{v\in V}\overline{K}(v)=2\pi\chi(M)$, there exists a PL metric
$\overline{d}$, unique up to scaling and isometry homotopic to the identity on
$(S,V)$, such that $\overline{d}$ is discrete conformal to $d$ and the
discrete curvature of $\overline{d}$ is $\overline{K}$.

The classical discrete Gaussian curvature defined by (1) is not a proper
discretization of the smooth Gaussian curvature on surfaces, which is
supported by the discussions in [4, 17]. To overcome the disadvantages of the
classical discrete Gaussian curvature, Ge and the first author [17] introduced
the combinatorial $\alpha$-curvature for Thurston’s Euclidean circle packing
metrics on surfaces. After that, there are lots of research activities on
combinatorial $\alpha$-curvature on surfaces and $3$-manifolds. See, for
instance, [9, 13, 14, 15, 16, 37, 38, 40, 43, 44] and others. Following [17],
the first author [40] introduced the following combinatorial
$\alpha$-curvature for vertex scaling of PL metrics on triangulated surfaces.

######  Definition 1.5 ([40]).

Suppose $(S,V,\mathcal{T})$ is a triangulated surface with a discrete PL
metric $l$, $\alpha\in\mathbb{R}$ is a constant and $u:V\rightarrow\mathbb{R}$
is a discrete conformal factor for $l$. The combinatorial $\alpha$-curvature
at $i\in V$ is defined to be

$R_{\alpha,i}=\frac{K_{i}}{e^{\alpha u_{i}}},$

where $K_{i}$ is the classical combinatorial curvature at $i\in V$ defined by
(1).

######  Remark 1.

If $\alpha=0$, the combinatorial $\alpha$-curvature $R_{\alpha}$ in Definition
1.5 is the classical discrete Gaussian curvature $K$. Taking $g_{i}=e^{u_{i}}$
as a discrete analogue of the smooth Riemannian metric, we have
$R_{\alpha,i}(\lambda g_{1},\cdots,\lambda
g_{V})=\lambda^{-\alpha}R_{\alpha,i}(g_{1},\cdots,g_{V})$ for any constant
$\lambda>0$. In the special case of $\alpha=1$, we have $R_{1,i}(\lambda
g_{1},\cdots,\lambda g_{V})=\lambda^{-1}R_{1,i}(g_{1},\cdots,g_{V})$, which is
parallelling to the transformation of smooth Gaussian curvature $K_{\lambda
g}=\lambda^{-1}K_{g}$ with $g$ being the Riemannian metric.

By the discrete Gauss-Bonnet formula (2) for the classical discrete Gaussian
curvature $K$, the combinatorial $\alpha$-curvature $R_{\alpha}$ in Definition
1.5 satisfies the following discrete Gauss-Bonnet formula $\sum_{i\in
V}R_{\alpha,i}e^{\alpha u_{i}}=2\pi\chi(S).$ Therefore, if
$\overline{R}\in\mathbb{R}^{V}$ is the combinatorial $\alpha$-curvature of
some discrete PL metric discrete conformal to $l$ on $(S,V,\mathcal{T})$ with
conformal factor $u$, then

$\sum_{i\in V}\overline{R}_{i}e^{\alpha u_{i}}=2\pi\chi(S),$ (3)

which is a discrete analogue of the constraint equation
$\int_{S}Ke^{2u}dV=2\pi\chi(S)$ in the smooth case [24, 3]. Following Kazdan-
Warner’s arguments in [24], the constraint equation (3) imposes the following
sign conditions on $\overline{R}$ depending on $\chi(S)$:

(a)

$\chi(M)>0$: $\overline{R}$ is positive somewhere,

(b)

$\chi(M)=0$: $\overline{R}$ changes sign (unless $\overline{R}\equiv 0$),

(c)

$\chi(M)<0$: $\overline{R}$ is negative somewhere.

It is natural to ask the following discrete version of Kazdan-Warner’s
question for combinatorial $\alpha$-curvature.

######  Question 1.6 (Discrete Kazdan-Warner Question).

Suppose $(S,V)$ is a marked surface with a PL metric $d$. Are the sign
conditions, depending on $\chi(S)$, sufficient conditions for a function
$\overline{R}$ defined on $V$ to be the combinatorial $\alpha$-curvature of
some polyhedral metric $d^{\prime}$ discrete conformal to $d$?

We prove the following discrete Kazdan-Warner type theorem for Discrete
Kazdan-Warner Question 1.6.

######  Theorem 1.7.

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$,
$\alpha\in\mathbb{R}$ is a constant and $\overline{R}$ is a given function
defined on $V$. Then there exists a PL metric with combinatorial
$\alpha$-curvature $\overline{R}$ in the discrete conformal class
$\mathcal{D}(d)$ if one of the following conditions is satisfied :

(1)

$\chi(S)>0,\ \alpha<0,\ \overline{R}>0$;

(2)

$\chi(S)<0,\ \alpha\neq 0,\ \overline{R}\leq 0,\ \overline{R}\not\equiv 0$;

(3)

$\chi(S)=0,\ \alpha\neq 0,\ \overline{R}\equiv 0$;

(4)

$\alpha=0$, $\overline{R}\in(-\infty,2\pi)$, $\sum_{i\in
V}\overline{R}_{i}=2\pi\chi(S)$.

######  Remark 2.

If $\overline{R}$ is a constant and $\alpha\overline{R}\leq 0$, the existence
of PL metric with combinatorial $\alpha$-curvature $\overline{R}$ in the
discrete conformal class $\mathcal{D}(d)$ has been proved in [40]. In the case
of $\alpha\overline{R}\leq 0$, the uniqueness of PL metric with combinatorial
$\alpha$-curvature $\overline{R}$ in the discrete conformal class
$\mathcal{D}(d)$ has been proved in [19, 40]. For the case
$\alpha\overline{R}>0$, the uniqueness is unknown. By the relationship of
combinatorial $\alpha$-curvature and the classical discrete Gaussian
curvature, the cases (3) and (4) in Theorem 1.7 are covered by Gu-Luo-Sun-Wu
[19]. Therefore, we just need to prove the cases (1) and (2) of Theorem 1.7.

The main tools for the proof of Theorem 1.7 are Gu-Luo-Sun-Wu’s discrete
conformal theory for PL metrics on surfaces [19] and variational principles
with constraints. The main ideas of the paper come from reading of Gu-Luo-Sun-
Wu [19] and Kouřimská [23]. A hyperbolic version of Theorem 1.4 has been
proved by Gu-Guo-Luo-Sun-Wu [18], which perfectly solves the classical
prescribing discerete Gaussian curvature problem in the hyperbolic background
geometry. For prescribing combinatorial $\alpha$-curvature problem in the
hyperbolic background geometry, the authors [43] recently obtained a
hyperbolic version of Theorem 1.7 using Luo’s combinatorial Yamabe flow and
Gu-Guo-Luo-Sun-Wu’s discrete conformal theory for piecewise hyperbolic metrics
on surfaces [18].

###  1.3 Organization of the paper

The paper is organized as follows. In Section 2, we recall the variational
principle introduced by Luo [26] for vertex scaling and Bobenko-Pinkall-
Spingborn’s development of Luo’s variational principle, and then recall the
discrete conformal theory established by Gu-Luo-Sun-Wu [19]. In Section 3, we
translate Theorem 1.7 into an optimization problem with constraints. In
Section 4, we prove Theorem 1.7.

Acknowledgements
The first author thanks Professor Feng Luo, Dr. Tianqi Wu, Dr. Yanwen Luo, Dr.
Wai Yeung Lam and Dr. Xiaoping Zhu for valuable communications. The authors
thank the organizers of FRG Workshop on Geometric Methods for Analyzing
Discrete Shapes for hosting the workshop that leads to this paper. The
research of the first author is supported by the Fundamental Research Funds
for the Central Universities under grant no. 2042020kf0199.

##  2 Discrete conformality of PL metrics on surfaces

In this section, we recall some facts about discrete conformality of PL
metrics that we need to use in the proof of Theorem 1.7.

###  2.1 The energy functions

Suppose $\\{ijk\\}\in F_{\mathcal{T}}$ is a triangle and
$l:E_{\mathcal{T}}\rightarrow\mathbb{R}_{>0}$ is a discrete PL metric on
$(S,V,\mathcal{T})$. Denote the inner angle in the triangle $\\{ijk\\}$ at the
vertex $i$ as $\theta_{i}$. Luo [26] proved the following result.

######  Lemma 2.1 ([26]).

Suppose $\\{ijk\\}\in F_{\mathcal{T}}$ is a triangle and
$l:E_{\mathcal{T}}\rightarrow\mathbb{R}_{>0}$ is a discrete PL metric on
$(S,V,\mathcal{T})$.

(1)

The admissible space of the discrete conformal factors for a triangle
$\\{ijk\\}\in F_{\mathcal{T}}$

$\displaystyle\Omega_{ijk}=\\{(u_{i},u_{j},u_{k})\in\mathbb{R}^{3}|\widetilde{l}_{ij},\widetilde{l}_{ik},\widetilde{l}_{jk}\text{
satisfy the triangle inequality}\\}$

is simply connected.

(2)

The Jacobian matrix
$\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is symmetric and negative semi-definite with kernel
$\\{t(1,1,1)^{T}|t\in\mathbb{R}\\}$ for any discrete conformal factor
$(u_{i},u_{j},u_{k})\in\Omega_{ijk}$.

Based on Lemma 2.1, Luo [26] constructed the following energy function for a
triangle $\\{ijk\\}\in F_{\mathcal{T}}$

$\displaystyle
F_{ijk}(u_{i},u_{j},u_{k})=-\int_{(0,0,0)}^{(u_{i},u_{j},u_{k})}\theta_{i}du_{i}+\theta_{j}du_{j}+\theta_{k}du_{k},$
(4)

which is a well-defined smooth locally convex function defined on
$\Omega_{ijk}$ with $\nabla F_{ijk}=(-\theta_{i},-\theta_{j},-\theta_{k})$ and

$\displaystyle
F_{ijk}((u_{i},u_{j},u_{k})+t(1,1,1))=F_{ijk}(u_{i},u_{j},u_{k})-\pi t$ (5)

for $t\in\mathbb{R}$. Motivated by [8], Luo [26] further introduced the
following energy function

$\displaystyle\mathbb{F}_{\mathcal{T}}(u)=\int_{0}^{u}\sum_{i\in
V}K_{i}du_{i}=\sum_{\\{ijk\\}\in
F_{\mathcal{T}}}F_{ijk}(u_{i},u_{j},u_{k})+2\pi\sum_{i\in V}u_{i},$ (6)

which is a locally convex smooth function of the discrete conformal factors
$u\in\cap_{\\{ijk\\}\in F_{\mathcal{T}}}\Omega_{ijk}$ with
$\nabla\mathbb{F}_{\mathcal{T}}(u)=K$. By (5) and the Gauss-Bonnet formula,
the energy function $\mathbb{F}_{\mathcal{T}}(u)$ has the following property

$\mathbb{F}_{\mathcal{T}}(u+c(1,\cdots,1))=\mathbb{F}_{\mathcal{T}}(u)+2c\pi\chi(S)$
(7)

for $c\in\mathbb{R}$ [26].

In the following of the paper, we need to use the explicit expression of
$\mathbb{F}_{\mathcal{T}}(u)$, which was first constructed up to a constant by
Bobenko-Pinkall-Spingborn [4] as follows. Set

$\Omega=\\{(x,y,z)\in\mathbb{R}^{3}|\
e^{x}+e^{y}>e^{z},e^{y}+e^{z}>e^{x},e^{z}+e^{x}>e^{y}\\}.$

For $(x,y,z)\in\Omega$, $e^{x},e^{y},e^{z}$ form the edge lengths of a
Euclidean triangle. Denote the inner angles facing $e^{x},e^{y},e^{z}$ as
$\alpha,\beta,\gamma$ respectively and set

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

to be Milnor’s Lobachevsky function [29]. Bobenko-Pinkall-Spingborn [4] then
defined the following function

$\displaystyle f:\Omega$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle(x,y,z)$ $\displaystyle\mapsto f(x,y,z)=\alpha x+\beta y+\gamma
z+\mathbb{L}(\alpha)+\mathbb{L}(\beta)+\mathbb{L}(\gamma).$

Using $f$ as building blocks, Bobenko-Pinkall-Spingborn [4] further
constructed the following function

$\displaystyle\mathbb{E}_{\mathcal{T}}(u)=\sum_{\\{ijk\\}\in
F_{\mathcal{T}}}\left(2f(\frac{\widetilde{\lambda}_{ij}}{2},\frac{\widetilde{\lambda}_{jk}}{2},\frac{\widetilde{\lambda}_{ki}}{2})-\frac{\pi}{2}(\widetilde{\lambda}_{ij}+\widetilde{\lambda}_{jk}+\widetilde{\lambda}_{ki})\right)+2\pi\sum_{i\in
V}u_{i}$ (8)

for $u\in\cap_{\\{ijk\\}\in F_{\mathcal{T}}}\Omega_{ijk}$, where
$\widetilde{\lambda}_{ij}$ is the logarithm length of
$\widetilde{l}_{ij}=l_{ij}e^{\frac{u_{i}+u_{j}}{2}}$. Bobenko-Pinkall-
Springborn [4] proved that $\nabla\mathbb{E}_{\mathcal{T}}=K$, which implies
that $\mathbb{E}_{\mathcal{T}}(u)$ and $\mathbb{F}_{\mathcal{T}}(u)$ differs
by some constant.

Bobenko-Pinkall-Springborn’s construction of $\mathbb{E}_{\mathcal{T}}(u)$ in
[4] is very elegant, but a little mysterious. It seems that Bobenko-Pinkall-
Springborn’s construction comes from their observation on the relationships of
vertex scaling and 3-dimensional hyperbolic geometry and can be taken as a
consequence of the Schläfli formula [32]. Once one has the explicit form of
the function $\mathbb{E}_{\mathcal{T}}(u)$, it is easy to check that
$\nabla\mathbb{E}_{\mathcal{T}}=K$. However, the construction of
$\mathbb{E}_{\mathcal{T}}(u)$ is not so easy. As Luo’s construction of the
energy function $\mathbb{F}_{\mathcal{T}}(u)$ in [26] is relatively direct and
easy, a natural question of independent interest is whether one can derive an
explicit form of $\mathbb{F}_{\mathcal{T}}(u)$ directly, which differs from
the explicit form of $\mathbb{E}_{\mathcal{T}}(u)$ by a constant. In the
following, we give such an argument, the idea of which comes from Yuhao Xue
from summer school 2017 at Tsinghua University. In fact, we just need to
derive an explicit form of the energy function
$\mathbb{F}_{ijk}(u_{i},u_{j},u_{k})$ for a triangle $\\{ijk\\}\in
F_{\mathcal{T}}$.

######  Proposition 2.2.

Suppose $\widetilde{l}$ is vertex scaling of $l$ on the triangle $\\{ijk\\}\in
F_{\mathcal{T}}$ with a conformal factor $(u_{i},u_{j},u_{k})\in\Omega_{ijk}$.
Denote the inner angle of the triangle with edge lengths
$l_{ij},l_{ik},l_{jk}$ as
$\overline{\Theta}_{i},\overline{\Theta}_{j},\overline{\Theta}_{k}$
respectively and denote the inner angle of the triangle with edge lengths
$\widetilde{l}_{ij},\widetilde{l}_{ik},\widetilde{l}_{jk}$ as
$\Theta_{i},\Theta_{j},\Theta_{k}$ respectively. Then the energy function
$F_{ijk}(u_{i},u_{j},u_{k})$ defined by (4) has the following explicit form

$\displaystyle F_{ijk}(u_{i},u_{j},u_{k})$ $\displaystyle=$
$\displaystyle-(\Theta_{i}u_{i}+\Theta_{j}u_{j}+\Theta_{k}u_{k})$
$\displaystyle+2\mathbb{L}(\Theta_{i})+2\mathbb{L}(\Theta_{j})+2\mathbb{L}(\Theta_{k})+2\Theta_{i}\ln
l_{jk}+2\Theta_{j}\ln l_{ik}+2\Theta_{k}\ln l_{ij}$
$\displaystyle-2\mathbb{L}(\overline{\Theta}_{i})-2\mathbb{L}(\overline{\Theta}_{j})-2\mathbb{L}(\overline{\Theta}_{k})-2\overline{\Theta}_{k}\ln
l_{ij}-2\overline{\Theta}_{j}\ln l_{ik}-2\overline{\Theta}_{i}\ln l_{jk}.$

###### Proof.

As $\Omega_{ijk}$ is simply connected by Lemma 2.1 and
$(0,0,0),(u_{i},u_{j},u_{k})\in\Omega_{ijk}$, we can assume that
$\gamma:[0,1]\rightarrow\Omega_{ijk}$ is a smooth path from $(0,0,0)$ to
$(u_{i},u_{j},u_{k})$. Denote the corresponding quantities along the path
$\gamma(t)$ as $u_{i}(t),\theta_{i}(t),l_{ij}(t)$ et al. Then
$u_{i}(1)=u_{i},u_{i}(0)=0,\theta_{i}(1)=\Theta_{i},\theta_{i}(0)=\overline{\Theta}_{i}$,
$l_{ij}(1)=\widetilde{l}_{ij}$, $l_{ij}(0)=l_{ij}$. By Definition 1.2, one can
solve $u_{i}(t),u_{j}(t),u_{k}(t)$ as follows

$\displaystyle
u_{i}(t)=\ln\frac{l_{ij}(t)l_{ik}(t)l_{jk}(0)}{l_{ij}(0)l_{ik}(0)l_{jk}(t)},u_{j}(t)=\ln\frac{l_{ij}(t)l_{ik}(0)l_{jk}(t)}{l_{ij}(0)l_{ik}(t)l_{jk}(0)},u_{k}(t)=\ln\frac{l_{ij}(0)l_{ik}(t)l_{jk}(t)}{l_{ij}(t)l_{ik}(0)l_{jk}(0)}.$
(9)

Suppose the circumcircle radius of the triangle with edge lengths
$l_{ij}(t),l_{ik}(t),l_{jk}(t)$ is $R(t)$, then

$\displaystyle
l_{ij}(t)=2R(t)\sin\theta_{k}(t),l_{ik}(t)=2R(t)\sin\theta_{j}(t),l_{jk}(t)=2R(t)\sin\theta_{i}(t).$
(10)

Submitting (10) into (9) gives

$\displaystyle u_{i}(t)=$ $\displaystyle 3\ln
R(t)+\ln\sin\theta_{j}(t)+\ln\sin\theta_{k}(t)-\ln\sin\theta_{i}(t)-\ln
l_{ij}(0)-\ln l_{ik}(0)+\ln l_{jk}(0),$ (11) $\displaystyle u_{j}(t)=$
$\displaystyle 3\ln
R(t)+\ln\sin\theta_{i}(t)+\ln\sin\theta_{k}(t)-\ln\sin\theta_{j}(t)-\ln
l_{ij}(0)-\ln l_{jk}(0)+\ln l_{ik}(0),$ $\displaystyle u_{k}(t)=$
$\displaystyle 3\ln
R(t)+\ln\sin\theta_{i}(t)+\ln\sin\theta_{j}(t)-\ln\sin\theta_{k}(t)-\ln
l_{ik}(0)-\ln l_{jk}(0)+\ln l_{ij}(0).$

By the definition of $F_{ijk}(u_{i},u_{j},u_{k})$ in (4), we have

$\displaystyle F_{ijk}(u_{i},u_{j},u_{k})$ (12) $\displaystyle=$
$\displaystyle-\int_{0}^{1}\theta_{i}(t)du_{i}(t)+\theta_{j}(t)du_{j}(t)+\theta_{k}(t)du_{k}(t)$
$\displaystyle=$
$\displaystyle-[\theta_{i}(t)u_{i}(t)+\theta_{j}(t)u_{j}(t)+\theta_{k}(t)u_{k}(t)]|^{1}_{0}+\int_{0}^{1}u_{i}(t)d\theta_{i}(t)+u_{j}(t)d\theta_{j}(t)+u_{k}(t)d\theta_{k}(t)$

by integration by parts. Submitting (11) into (12) gives

$\displaystyle F_{ijk}(u_{i},u_{j},u_{k})$ $\displaystyle=$
$\displaystyle-[\theta_{i}(t)u_{i}(t)+\theta_{j}(t)u_{j}(t)+\theta_{k}(t)u_{k}(t)]|^{1}_{0}$
$\displaystyle-2\int_{0}^{1}[\theta_{i}^{\prime}(t)\ln\sin\theta_{i}(t)+\theta_{j}^{\prime}(t)\ln\sin\theta_{j}(t)+\theta_{k}^{\prime}(t)\ln\sin\theta_{k}(t)]dt$
$\displaystyle+2\int_{0}^{1}[\ln l_{ij}(0)\theta_{k}^{\prime}(t)+\ln
l_{ik}(0)\theta_{j}^{\prime}(t)+\ln l_{jk}(0)\theta_{i}^{\prime}(t)]dt$
$\displaystyle=$
$\displaystyle-(\Theta_{i}u_{i}+\Theta_{j}u_{j}+\Theta_{k}u_{k})$
$\displaystyle+2\mathbb{L}(\Theta_{i})+2\mathbb{L}(\Theta_{j})+2\mathbb{L}(\Theta_{k})-2\mathbb{L}(\overline{\Theta}_{i})-2\mathbb{L}(\overline{\Theta}_{j})-2\mathbb{L}(\overline{\Theta}_{k})$
$\displaystyle+2\ln l_{ij}(0)(\Theta_{k}-\overline{\Theta}_{k})+2\ln
l_{ik}(0)(\Theta_{j}-\overline{\Theta}_{j})+2\ln
l_{jk}(0)(\Theta_{i}-\overline{\Theta}_{i}),$

where the identity
$\theta_{i}^{\prime}(t)+\theta_{j}^{\prime}(t)+\theta_{k}^{\prime}(t)\equiv 0$
is used in the first equality. ∎

######  Remark 3.

Set

$\displaystyle
h(u_{i},u_{j},u_{j})=2f(\frac{\widetilde{\lambda}_{ij}}{2},\frac{\widetilde{\lambda}_{jk}}{2},\frac{\widetilde{\lambda}_{ki}}{2})-\frac{\pi}{2}(\widetilde{\lambda}_{ij}+\widetilde{\lambda}_{jk}+\widetilde{\lambda}_{ki}),$

which is a building block of $\mathbb{E}_{\mathcal{T}}(u)$ in (8). By direct
calculations, we have

$\displaystyle h(u_{i},u_{j},u_{j})=$
$\displaystyle-(\Theta_{i}u_{i}+\Theta_{j}u_{j}+\Theta_{k}u_{k})$
$\displaystyle+2\mathbb{L}(\Theta_{i})+2\mathbb{L}(\Theta_{j})+2\mathbb{L}(\Theta_{k})+2\Theta_{i}\ln
l_{jk}+2\Theta_{j}\ln l_{ik}+2\Theta_{k}\ln l_{ij}$ $\displaystyle-\pi(\ln
l_{ij}+\ln l_{ik}+\ln l_{jk}),$

which implies that $F_{ijk}(u_{i},u_{j},u_{k})$ differ from
$h(u_{i},u_{j},u_{j})$ by a constant depending on the background discrete PL
metric $l$. As a direct consequence, Luo’s energy function
$\mathbb{F}_{\mathcal{T}}(u)$ in (6) differs from Bobenko-Pinkall-Springborn’s
energy function $\mathbb{E}_{\mathcal{T}}(u)$ in (8) by a constant depending
on the background discrete PL metric $l$.

######  Remark 4.

There are two approaches to extend the energy function
$\mathbb{F}_{\mathcal{T}}(u)$ or $\mathbb{E}_{\mathcal{T}}(u)$ to be a convex
function defined on $\mathbb{R}^{V}$ introduced by Bobenko-Pinkall-Springborn
[4] and Gu-Luo-Sun-Wu [19] independently. In [4], the function
$\mathbb{E}_{\mathcal{T}}(u)$ is extended to be a $C^{1}$ smooth convex
function defined on $\mathbb{R}^{V}$ by exploiting the fact that the function
$f$ defined on $\Omega$ could be extended to a $C^{1}$ smooth convex function
on $\mathbb{R}^{3}$ by extending the inner angles of a triangle by constants.
Using Luo’ development [27] of Bobenko-Pinkall-Spingborn’s extension [4], Ge-
Jiang [11] similarly extended the potential function
$\mathbb{F}_{\mathcal{T}}(u)$ defined by (6) to be a $C^{1}$-smooth convex
function defined on $\mathbb{R}^{V}$. Luo’s development [27] of Bobenko-
Pinkall-Spingborn’s extension [4] has recently been further developed to
handle other cases. See, for instance, [38, 39, 41, 42] and others. In [19],
the function $\mathbb{F}_{\mathcal{T}}(u)$ is extended to be a $C^{2}$-smooth
convex function defined on $\mathbb{R}^{V}$ by changing the triangulation of
the marked surface under the Delaunay condition, based on which Gu-Luo-Sun-Wu
proved the Discrete Uniformization Theorem 1.4 for PL metrics on closed
surfaces. The first approach could not ensure the triangles being non-
degenerate, while the second approach could.

###  2.2 Gu-Luo-Sun-Wu’s discrete conformal theory

Based on Penner’s decorated Teichimüller space theory [31], Gu-Luo-Sun-Wu [19]
established the discrete conformal theory for PL metrics on compact surfaces
and proved Theorem 1.4. In the following, we recall some results in [19] that
we need. We will not formally involve too much notions and results on
decorated Teichimüller space. For more details of these results, please refer
to Gu-Luo-Sun-Wu’s original work [19].

######  Theorem 2.3 ([19] Corollary 4.7).

Suppose $d$ is a PL metric on the marked surface $(S,V)$. Then there exists a
$C^{1}$ diffeomorphism $\phi:\mathcal{D}(d)\rightarrow\mathbb{R}^{V}$.

By Theorem 2.3, the discrete conformal class $\mathcal{D}(d)$ is parameterized
by $\mathbb{R}^{V}$. For simplicity, for any $d^{\prime}\in\mathcal{D}(d)$, we
denote it by $d(u)$ for some $u\in\mathbb{R}^{V}$. Suppose $\mathcal{T}$ is a
triangulation of the marked surface $(S,V)$. Set

$\mathcal{A}_{\mathcal{T}}=\\{u\in R^{V}|\ \mathcal{T}\ \text{is isotopic to a
Delaunay triangulation of}\ (S,V,d(u))\\}.$

Based on Akiyoshi’s finiteness theorem in [1] (see also Appendix in [19] for a
new proof), Gu-Luo-Sun-Wu [19] proved the following result.

######  Theorem 2.4 ([19] Lemma 5.1).

Let

$J=\\{\mathcal{T}|\mathcal{A}_{\mathcal{T}}\ \text{has nonempty interior in}\
\mathbb{R}^{V}\\}.$

Then $J$ is a finite set, $\mathbb{R}^{V}=\cup_{\mathcal{T}_{i}\in
J}\mathcal{A}_{\mathcal{T}_{i}}$ and $\mathcal{A}_{\mathcal{T}_{i}}$ is real
analytically diffeomorphic to a closed convex polytope in $\mathbb{R}^{V}$.

By Theorem 2.3, Gu-Luo-Sun-Wu [19] introduced the following globally defined
$C^{1}$ smooth combinatorial curvature function

$\displaystyle\mathbf{F}:\mathbb{R}^{V}$
$\displaystyle\rightarrow(-\infty,2\pi)^{V}$ $\displaystyle u$
$\displaystyle\mapsto K(d(u)).$

Combining with Lemma 2.1, Gu-Luo-Sun-Wu [19] further constructed a globally
defined energy function with $\mathbf{F}$ as gradient, which plays an
important role in the proof of Theorem 1.4.

######  Theorem 2.5 ([19] Proposition 5.2).

There exists a $C^{2}$-smooth convex function

$\displaystyle\mathbb{E}:\mathbb{R}^{V}$ $\displaystyle\rightarrow\mathbb{R}$
(13) $\displaystyle u$ $\displaystyle\mapsto\int_{0}^{u}\sum_{i\in
V}\mathbf{F}_{i}du_{i}$

so that its gradient $\nabla\mathbb{E}=\mathbf{F}$ and the restriction of
$\mathbb{E}$ to the hyperplane $\\{u\in\mathbb{R}^{V}|\sum_{i\in V}u_{i}=0\\}$
is strictly convex.

######  Remark 5.

By the construction in Theorem 2.5, for $\mathcal{T}\in J$, the restriction
$\mathbb{E}|_{\mathcal{A}_{\mathcal{T}}}$ differs from
$\mathbb{F}_{\mathcal{T}}$ in (6) by a constant. As a consequence of (7),
$\mathbb{E}$ has the following property

$\mathbb{E}(u+c(1,\cdots,1))=\mathbb{E}(u)+2c\pi\chi(S)$ (14)

for any $c\in\mathbb{R}$.

##  3 Variational principles with constraints

In this section, we translate Theorem 1.7 into an optimization problem with
inequality constraints by variational principles, which involve the function
$\mathbb{E}$ defined in (13).

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$,
$\alpha\in\mathbb{R}$ is a non-zero constant and $\overline{R}$ is a given
function defined on $V$. Set

$\mathcal{A}=\\{u\in\mathbb{R}^{V}|0>\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}\geq 2\pi\chi(S),\ \overline{R}\leq 0,\ \overline{R}\not\equiv 0\\},$
(15) $\mathcal{B}=\\{u\in\mathbb{R}^{V}|0<\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}\leq 2\pi\chi(S),\ \overline{R}>0\\},$ (16)
$\mathcal{C}=\\{u\in\mathbb{R}^{V}|\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}\leq 2\pi\chi(S)<0,\ \overline{R}\leq 0,\ \overline{R}\not\equiv 0\\}.$
(17)

######  Proposition 3.1.

The sets $\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ are unbounded closed
subsets of $\mathbb{R}^{V}$.

###### Proof.

It is obvious that the sets $\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ are
closed subsets of $\mathbb{R}^{V}$. For the set $\mathcal{A}$, by direct
calculations,

$\sum_{i\in V}\overline{R}_{i}e^{\alpha(u_{i}+c)}=e^{\alpha c}\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}\geq 2\pi\chi(S)$

is equivalent to

$c\geq\frac{1}{\alpha}\log\frac{2\pi\chi(S)}{\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}}$

under the condition $\alpha<0$;

$\sum_{i\in V}\overline{R}_{i}e^{\alpha(u_{i}+c)}=e^{\alpha c}\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}\geq 2\pi\chi(S)$

is equivalent to

$c\leq\frac{1}{\alpha}\log\frac{2\pi\chi(S)}{\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}}$

under the condition $\alpha>0$. Therefore, the set $\mathcal{A}$ is unbounded.
Similarly, the sets $\mathcal{B}$ and $\mathcal{C}$ are unbounded. ∎

According to Proposition 3.1, we have following result.

######  Lemma 3.2.

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$,
$\alpha\in\mathbb{R}$ is a constant and $\overline{R}$ is a given function
defined on $V$. If one of the following three conditions is satisfied

(1)

$\alpha>0$ and the energy function $\mathbb{E}$ attains a minimum in the set
$\mathcal{A}$,

(2)

$\alpha<0$ and the energy function $\mathbb{E}$ attains a minimum in the set
$\mathcal{B}$,

(3)

$\alpha<0$ and the energy function $\mathbb{E}$ attains a minimum in the set
$\mathcal{C}$,

then the minimum value point of $\mathbb{E}$ lies in the set
$\\{u\in\mathbb{R}^{V}|\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}=2\pi\chi(S)\\}$.

###### Proof.

Let $\alpha>0$ and suppose the function $\mathbb{E}$ attains a minimum at
$u\in\mathcal{A}$. The definition of $\mathcal{A}$ in (15) implies
$\chi(S)<0$. Taking $c_{0}=\frac{1}{\alpha}\log\frac{2\pi\chi(S)}{\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}}$, then $c_{0}\geq 0$. By the proof of
Proposition 3.1, $u+c_{0}\mathbb{I}\in\mathcal{A}$. Therefore, by the additive
property of the function $\mathbb{E}$ in (14), we have

$\mathbb{E}(u)\leq\mathbb{E}(u+c_{0}\mathbb{I})=\mathbb{E}(u)+2\pi
c_{0}\chi(S),$

which implies $c_{0}\leq 0$ by $\chi(S)<0$. Hence $c_{0}=0$ and $\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}=2\pi\chi(S)$. This proves the case of
$\mathbf{(1)}$. The proofs for the cases $\mathbf{(2)}$ and $\mathbf{(3)}$ are
similar, we omit the details here. ∎

By Lemma 3.2, we translate Theorem 1.7 into the following theorem, which is a
non-convex optimization problem with inequality constraints.

######  Theorem 3.3.

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$ and $\chi(S)\neq
0$, $\alpha\in\mathbb{R}$ is a non-zero constant and $\overline{R}$ is a given
function defined on $V$.

(1)

If $\alpha>0$ and the energy function $\mathbb{E}$ attains a minimum in
$\mathcal{A}$, then there exists a PL metric in the conformal class
$\mathcal{D}(d)$ with combinatorial $\alpha$-curvature $\overline{R}\leq 0$
and $\overline{R}\not\equiv 0$;

(2)

If $\alpha<0$ and the energy function $\mathbb{E}$ attains a minimum in
$\mathcal{B}$, then there exists a PL metric in the conformal class
$\mathcal{D}(d)$ with combinatorial $\alpha$-curvature $\overline{R}>0$;

(3)

If $\alpha<0$ and the energy function $\mathbb{E}$ attains a minimum in
$\mathcal{C}$, then there exists a PL metric in the conformal class
$\mathcal{D}(d)$ with combinatorial $\alpha$-curvature $\overline{R}\leq 0$
and $\overline{R}\not\equiv 0$.

###### Proof.

Lemma 3.2 shows that if $u\in\mathbb{R}^{V}$ is a minimum of the energy
function $\mathbb{E}$ defined on one of these sets, then $\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}=2\pi\chi(S)$. The conclusion follows from
the following claim.
Claim : Up to scaling, the PL-metrics with prescribed combinatorial
$\alpha$-curvature in the discrete conformal class $\mathcal{D}(d)$ are in
one-to-one correspondence with the critical points of the function
$\mathbb{E}$ under the constraint $\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}=2\pi\chi(S)$.

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

$H(u,\lambda)=\mathbb{E}(u)+\lambda\left(\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}-2\pi\chi(S)\right),$

where $\lambda\in\mathbb{R}$ is a Lagrange multiplier. If $u$ is a critical
point of the function $\mathbb{E}$ under the constraint $\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}=2\pi\chi(S)$, then

$0=\frac{\partial H(u,\lambda)}{\partial
u_{i}}=K_{i}+\lambda\alpha\overline{R}_{i}e^{\alpha u_{i}},$

which implies

$R_{\alpha,i}=\frac{K_{i}}{e^{\alpha u_{i}}}=-\lambda\alpha\overline{R}_{i}.$

By the discrete Gauss-Bonnet formula (2), the Lagrange multiplier $\lambda$
satisfies

$\lambda=-\frac{2\pi\chi(S)}{\alpha\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}}=-\frac{1}{\alpha}$

under the constraint $\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}=2\pi\chi(S)$, which implies the combinatorial $\alpha$-curvature

$R_{\alpha,i}=-\lambda\alpha\overline{R}_{i}=\frac{2\pi\chi(S)}{\sum_{i\in
V}\overline{R}_{i}e^{\alpha u_{i}}}\overline{R}_{i}=\overline{R}_{i}$

under the constraint $\sum_{i\in V}\overline{R}_{i}e^{\alpha
u_{i}}=2\pi\chi(S)$. ∎

##  4 Proof of Theorem 1.7

In this section, we give a proof for Theorem 1.7. By Theorem 3.3, we just need
to prove that the energy function $\mathbb{E}$ attains the minimum in the sets
$\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$. Recall the following classical
result from calculus.

######  Theorem 4.1.

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
$\\{x_{n_{k}}\\}_{k\in\mathbb{N}}$ such that
$\lim_{k\rightarrow+\infty}f(x_{n_{k}})=+\infty$, then $f$ attains a minimum
in $A$.

One can refer to [22] (Section 4.1) for a proof of Theorem 4.1. The majority
of the conditions in Theorem 4.1 are satisfied, including the sets
$\mathcal{A},\ \mathcal{B}$ and $\mathcal{C}$ are closed subsets of
$\mathbb{R}^{V}$ by Proposition 3.1 and the energy function $\mathbb{E}$ is
continuous by Theorem 2.5. To prove Theorem 1.7, we just need to prove the
following theorem by Theorem 3.3 and Theorem 4.1.

######  Theorem 4.2.

Suppose $(S,V,d)$ is a marked surface with a PL metric $d$,
$\alpha\in\mathbb{R}$ is a constant and $\overline{R}$ is a given function
defined on $V$. If one of the following three conditions is satisfied,

(1)

$\alpha>0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{A}$,

(2)

$\alpha<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{B}$,

(3)

$\alpha<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{C}$,

then there exist a subsequence $\\{u_{n_{k}}\\}_{k\in\mathbb{N}}$ of
$\\{u_{n}\\}_{n\in\mathbb{N}}$ such that
$\lim_{k\rightarrow+\infty}\mathbb{E}(u_{n_{k}})=+\infty$.

Let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be an unbounded sequence in
$\mathbb{R}^{V}$, denote its coordinate sequence at $j\in V$ by
$\\{u_{j,n}\\}_{n\in\mathbb{N}}$. Following [23], we use the following
convention.

######  Convention 4.3.

The sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ possesses the following
properties:

(1)

It lies in one cell $\mathcal{A}_{\mathcal{T}}$ of $\mathbb{R}^{V}$ for some
$\mathcal{T}\in J$ given by Theorem 2.4;

(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 converge,
diverge properly to $+\infty$, or diverges properly to $-\infty$;

(4)

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

By Theorem 2.4, it is obvious that every sequence in $\mathbb{R}^{V}$
possesses a subsequence satisfying these properties in Convention 4.3, hence
the sequence in Convention 4.3 could be chosen without loss of generality. To
prove Theorem 4.2, we further need the following two results obtained by
Kouřimská [23].

######  Lemma 4.4 ([23] Corollary 5.6).

In every triangle $\\{ijk\\}\in F_{\mathcal{T}}$, at least two of the three
sequences $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$,
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ and
$(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge.

######  Lemma 4.5 ([23] Lemma 5.11).

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

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

The proof of Lemma 4.5 is based on an interesting analysis of the explicit
form of the energy function $\mathbb{F}_{\mathcal{T}}$ or
$\mathbb{E}_{\mathcal{T}}$. Readers can refer to [23, 22] for the proof.

Proof of Theorem 4.2: Assume that $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an
unbounded sequence satisfying Convention 4.3. We just need to prove that
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$.

(1)

Let $\alpha>0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{A}$. The definition of $\mathcal{A}$ in (15) implies $\chi(S)<0$,
$\overline{R}\leq 0$ and $\overline{R}\not\equiv 0$. Since the sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ lies in $\mathcal{A}$, we have

$0>\sum_{j\in V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}=e^{-\alpha
u_{i^{*},n}}\cdot\sum_{j\in V}\overline{R}_{j}e^{\alpha u_{j,n}}\geq
2\pi\chi(S)e^{-\alpha u_{i^{*},n}}.$ (18)

Note that $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$
converges to a finite number or diverges properly to $+\infty$ by Convention
4.3.

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 $j\in V$ by
Convention 4.3, which implies $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ converges to a finite
negative number by $\overline{R}\leq 0$ and $\overline{R}\not\equiv 0$.
Therefore, $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ is bounded from above by (18),
$\alpha>0$ and $\chi(S)<0$, which implies $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$
converges to a finite number or diverges to $-\infty$. If
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ converges to a finite number, then
$\\{u_{j,n}\\}_{n\in\mathbb{N}}$ are bounded for all $j\in V$ by
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges 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 $-\infty$.
Combining this with $\chi(S)<0$ and Lemma 4.5, we have
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$.

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 $j\in V$ such
that the sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges properly
to $+\infty$. By Lemma 4.4, there exists at least one vertex $k\in V$ such
that the sequence $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges (for
example $(u_{i^{*},n}-u_{i^{*},n})_{n\in\mathbb{N}}$). Therefore,
$e^{\alpha(u_{j,n}-u_{i^{*},n})}$ converges to a finite positive number or
diverges properly to $+\infty$ and for at least one vertex $j\in V$ the term
$e^{\alpha(u_{j,n}-u_{i^{*},n})}$ converges to a finite positive number.
Combining $\overline{R}\leq 0$ and $\overline{R}\not\equiv 0$, $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ tends to either a finite
negative number or $-\infty$. If $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ tends to a finite negative
number, then $u_{i^{*},n}$ is bounded from above by (18), which implies that
$u_{i^{*},n}\chi(S)$ is bounded from below. Combining with the assumption that
$\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ diverges
properly to $+\infty$, we have
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$ by Lemma 4.5. If
$\sum_{j\in V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ tends to
$-\infty$, we have $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly to
$-\infty$ by (18), which implies
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$ by $\chi(S)<0$ and Lemma
4.5.

(2)

Let $\alpha<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{B}$. The definition of $\mathcal{B}$ in (16) implies $\chi(S)>0$ and
$\overline{R}>0$. Since the sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ lies in
$\mathcal{B}$, we have

$0<\sum_{j\in V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}=e^{-\alpha
u_{i^{*},n}}\cdot\sum_{j\in V}\overline{R}_{j}e^{\alpha u_{j,n}}\leq
2\pi\chi(S)e^{-\alpha u_{i^{*},n}}.$ (19)

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$
converges, then the sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$
converges for all $j\in V$ by Convention 4.3. Note that $\alpha<0$,
$\chi(S)>0$ and $\overline{R}>0$, we have $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$
is bounded from below by (19), which implies
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ converges to a finite number or diverges
properly to $+\infty$ by Convention 4.3. Combining this with
$\\{u_{n}\\}_{n\in\mathbb{N}}$ is unbounded and
$(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges for all $j\in V$, we have
the sequence $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly to
$+\infty$. As a result, we have
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$ by Lemma 4.5 and
$\chi(S)>0$.

If the sequence $\left(\sum_{j\in
V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ diverges properly to
$+\infty$, then there exists at least one vertex $j\in V$ such that the
sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges properly to
$+\infty$. By Lemma 4.4, there exists at least one vertex $k\in V$ such that
the sequence $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges (for example
$(u_{i^{*},n}-u_{i^{*},n})_{n\in\mathbb{N}}$). Therefore, $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ has a positive lower bound
by $\overline{R}>0$ and $\alpha<0$, which implies $2\pi\chi(S)e^{-\alpha
u_{i^{*},n}}$ has a positive lower bound by (19). Therefore,
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ is bounded from below by $\alpha<0$ and
$\chi(S)>0$ and hence $u_{i^{*},n}\chi(S)$ is bounded from below. Combining
this with $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$
diverges properly to $+\infty$, we have
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$ by Lemma 4.5.

(3)

Let $\alpha<0$ and $\\{u_{n}\\}_{n\in\mathbb{N}}$ is an unbounded sequence in
$\mathcal{C}$. The definition of $\mathcal{C}$ in (17) implies $\chi(S)<0$,
$\overline{R}\leq 0$ and $\overline{R}\not\equiv 0$. Since the sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}$ lies in $\mathcal{C}$, we have

$\sum_{j\in V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}=e^{-\alpha
u_{i^{*},n}}\cdot\sum_{j\in V}\overline{R}_{j}e^{\alpha u_{j,n}}\leq
2\pi\chi(S)e^{-\alpha u_{i^{*},n}}<0.$ (20)

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$
converges, then the sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$
converges for all $j\in V$ by Convention 4.3, which implies that $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ converges to a finite
negative number by $\overline{R}\leq 0$ and $\overline{R}\not\equiv 0$.
Combining this with $\alpha<0$ and $\chi(S)<0$, we have
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ is bounded from above by (20). As
$\\{u_{n}\\}_{n\in\mathbb{N}}$ is unbounded, then similar to the arguments
above, the sequence $\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly to
$-\infty$. Combining this with $\chi(S)<0$ and Lemma 4.5, we have
$\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$.

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 $j\in V$ such
that the sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges properly
to $+\infty$. By Lemma 4.4, there exists at least one vertex $k\in V$ such
that the sequence $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges (for
example $(u_{i^{*},n}-u_{i^{*},n})_{n\in\mathbb{N}}$). Therefore, $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ either tends to zero or to
a finite negative number by $\alpha<0$, $\overline{R}\leq 0$ and
$\overline{R}\not\equiv 0$. If $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ tends to zero, then
$2\pi\chi(S)e^{-\alpha u_{i^{*},n}}$ tends to zero by (20), which implies
$\\{u_{i^{*},n}\\}_{n\in\mathbb{N}}$ diverges properly to $-\infty$ by
$\alpha<0$ and $\chi(S)<0$. Combining this with $\chi(S)<0$ and Lemma 4.5, we
have $\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$. If $\sum_{j\in
V}\overline{R}_{j}e^{\alpha(u_{j,n}-u_{i^{*},n})}$ tends to a finite negative
number, we have $2\pi\chi(S)e^{-\alpha u_{i^{*},n}}$ has a negative lower
bound by (20), which implies $u_{i^{*},n}$ is bounded from above by $\alpha<0$
and $\chi(S)<0$. Combining this with $\chi(S)<0$ and $\left(\sum_{j\in
V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ diverges properly to
$+\infty$, we have $\lim_{n\rightarrow+\infty}\mathbb{E}(u_{n})=+\infty$ by
Lemma 4.5.

∎

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(Xu Xu) School of Mathematics and Statistics, Wuhan University, Wuhan 430072,
P.R. China

E-mail<EMAIL_ADDRESS>

(Chao Zheng) School of Mathematics and Statistics, Wuhan University, Wuhan
430072, P.R. China

E-mail<EMAIL_ADDRESS>


# Converter: A CEAML Reasoner Python package to Streamline Orchestration
Across Cloud and Edge Continuum

Ioannis Korontanis Antonios Makris, and Konstantinos Tserpes I. Korontanis, A.
Makris and K.Tserpes are with the Department of Informatics and Telematics,
Harokopio University of Athens, Greece and also with the School of Electrical
and Computer Engineering, National Technical University of Athens. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In recent years, there has been a concerted effort in both industry and
research sectors to innovate new approaches to DevOps. The primary aim is to
facilitate developers in transitioning their applications to Cloud or Edge
platforms utilizing Docker or Kubernetes. This paper presents a tool called
Converter, designed to interpret a TOSCA extension called CEAML and convert
the descriptions into Kubernetes or Kubevirt definition files. Converter is
available as a Python package and is recommended for use by orchestrators as
an auxiliary tool for implementing CEAML.

###### Index Terms:

Edge Computing, Cloud Computing, DevOps

##  I Introduction

Numerous efforts have been documented in the literature aiming to create or
extend cloud modeling languages capable of describing the behavior of deployed
microservices within a platform. Supporting tools are often provided alongside
the languages to facilitate their implementation. Achilleos et al. as
mentioned in [1], refer to tools that convert high-level cloud modeling
language descriptions into technical ones as reasoners.

A cloud modeling language that is quite popular in the research field is
TOSCA111https://docs.oasis-open.org/tosca/TOSCA-Simple-Profile-
YAML/v1.3/os/TOSCA-Simple-Profile-YAML-v1.3-os.html, developed by OASIS. TOSCA
aims to depict how multi-component applications are deployed across Cloud or
Edge resources in a way that’s not tied to any specific technology. Many
papers discuss methods for utilizing TOSCA or its extensions, in conjunction
with a reasoning tool, to facilitate application migration. For example,
Tosker[2] is a tool capable to translate models written with a TOSCA
extension, delineating Docker containers within Docker deployment plans.
Another case is CloudCAMP[3], which translates TOSCA into infrastructure as
code, which serves as deployment plans for the described applications in the
presented platform. A more sophisticated tool named MICADO Submitter is
presented by DesLaurier et al. [4]. MICADO Submitter is capable to translate
descriptions written in a TOSCA extended language into deployment plans to
Docker Swarm and Kubernetes. This tool was utilized by the orchestrator of the
system which applied the deployment plans. TOSCA can also facilitate
deployments that aren’t tied to containerization. A notable example is the
ARIA project, detailed at
222https://incubator.apache.org/projects/ariatosca.html, which is able to
convert TOSCA models into deployments for applications across various cloud
providers such as Microsoft Azure Cloud, Amazon Web Services, VMware, and
OpenStack.

Similarly, to streamline the implementation of our cloud modeling language,
CEAML, we developed a reasoner called Converter. CEAML is designed based on
the research conducted on [5]. It’s main purpose is to describe the deployment
and runtime adaptation of applications in Cloud and Edge environments. CEAML
is an extension of TOSCA designed to employ a more user-friendy syntax for
describing application components, resources and orchestrator behaviour based
on event-driven logic. This paper does not delve into the semantics of the
language; rather, it offers guidance on utilizing Converter to implement it
within a system. It’s important to clarify that the Converter alone does not
function as an orchestrator. However, it can be employed by an orchestrator to
execute deployment and runtime adjustments according to the descriptions
outlined in a CEAML model. The outputs of the Converter are configuration
files for Kubernetes or Kubevirt. Kubevirt333https://kubevirt.io/ is a plugin
that could be installed on Kubernetes to enable the deployment of virtual
machines.

The rest of the paper is organized as follows:

  * •

Section II delves into the functionalities Converter provides to an
orchestrator for executing specified tasks.

  * •

Section III investigates the importance of integrating Converter as a reasoner
and introduces a proposed pipeline

  * •

Section IV summarizes the conclusions drawn from the utilization of the
Converter reasoner.

##  II Package Functionalities

Converter is provided as a Python
package444https://pypi.org/project/converter-package/ and its source code is
available on Github555https://github.com/Efficient-Computing-Lab/Converter. To
generate the necessary outputs, Converter requires the appropriate CEAML model
(Figure 1). For an application described in CEAML, Converter is able to
generate multiple Kubernetes and Kubevirt configurations:

  * •

A namespace with a unique ID that would contain application components. The
unique ID is required to identify components that belong to the same
application and being deployed into different clusters.

  * •

Multiple deployment plans for Kubernetes, one for each application component.
These deployment plans are required for components that should be deployed as
pods.

  * •

Multiple secrets, one for each application component. Secrets are needed for
Kubernetes and Kubevirt in order to pull Docker or virtual machine images from
private registries.

  * •

Multiple persistent volumes for application components that need to have
persistent storage

  * •

Multiple load balancers for application components that need to have an
external IP

  * •

Multiple virtual machine deployments for each application component that need
to be deployed as a virtual machine on Kubevirt.

Figure 1: Converter input and outputs

From the aforementioned outputs and their variations, the Converter
facilitates three primary orchestration operations. These operations are
applicable to either a single Kubernetes or Kubevirt cluster, or across
multiple clusters:

  * •

Generate deployment plans for application components on Kubernetes or Kubevirt
(Section II-A).

  * •

Generate termination plans for application components that run on Kubernetes
or Kubevirt clusters (Section II-B).

  * •

Generate scale out plans for application components that run on Kubernetes or
Kubevirt clusters (Section II-C).

###  II-A Deployment Plans

To generate a deployment plan for a CEAML-described application on Kubernetes
or Kubevirt, developers must provide the following necessary inputs to the
Converter:

  1. 1.

Provide access tokens for the private registry housing their images and encode
them in base64.

  2. 2.

Specify the path of the model written in CEAML

  3. 3.

Indicate the version of their application

  4. 4.

Provide external IPs that could potentially be used by a component

  5. 5.

Specify the cluster ID to which the application should be deployed

  6. 6.

Provide a list of available GPUs on the cluster. If there are no GPUs
available, provide an empty list.

An example demonstrating how to provide inputs 1. and 2. is depicted in Figure
2. In this scenario, there’s a function called “selector” that requires the
application’s name as input. If the name matches one of the predefined ones,
it retrieves the access tokens and creates the base64-encoded version of the
access token. Additionally, the function defines the path of the model written
in CEAML. Finally, the function outputs the base64 access token, the
application’s name, and the model’s path.

Figure 2: Code snippet to provide access tokens and CEAML models

Figure 3 presents a way to provide inputs 3., 4., 5. and 6. The code snippet
presents a function named deployment that retrieves the outputs of the
previously mentioned selector function. The application version, cluster ID,
external IP and GPU list are hardcoded in order to simplify the example. In
any situation, you can also input these values through a mechanism instead of
embedding them directly into your code.

Figure 3: Code snippet for deployment plans

Combining the aforementioned figures provides a comprehensive example of
generating deployment files with Converter. Figure 3 demonstrates that
Converter requires several functions to be called for generating deployment
plans. The explanation of these functions are presented in Table I.

TABLE I: Description of deployment related functions Function |  Description
---|---
ID.generate_k3s_namespace, Converter.namespace |  Generate a namespace with a unique ID. The ID.generate_k3s_namespace function requires the application name, version, and a random alphanumeric. After generating the unique ID, call Converter.namespace and pass the unique ID as a parameter to create the K3s namespace. (Required)
MatchingModel.generate |  Generate a model for the orchestrator with hardware requirements of components. This function requires a list containing all instances of CEAML entities (nodeList) and the unique ID of the instance to be deployed (application_instance). (Optional)
Converter.secret_generation |  Generate required secrets. This function requires the base64 version of access tokens and the unique ID of the instance to be deployed. (Required)
Converter.tosca_to_k8s |  Generate deployment files, persistent volumes, and services. This function requires a list containing all instances of CEAML entities (nodeList), the unique ID of the instance to be deployed (application_instance), cluster ID, potential external IP, and potential GPU list. (Required)
ActionModel.generate |  Generate a model that includes components and related actions. This function requires a list containing all instances of CEAML entities (nodeList) and the unique ID of the instance to be deployed (application_instance). (Optional)
WorkflowModel.generate |  Generate a model that includes workflows with runtime adaptation actions to be performed under certain conditions. This function requires a list containing all instances of CEAML entities (nodeList) and the unique ID of the instance to be deployed (application_instance). (Optional)

In case developers want to use a conventional deployment on Kubernetes or
Kubevirt, they need to execute the required steps. However, in case they have
created a distributed orchestrator that operates across clusters, they may
also consider utilizing the optional steps (as indicated in Table I) to assist
in the orchestration process. Especially beneficial for distributed
orchestrators are the matchmaking model and the workflow model. The
matchmaking model will provide a submodel of CEAML that includes the described
application components and their hardware requirements, in order to assist
orchestrators to find the desirable hosts. On the other hand, the workflow
model contains instructions on how the orchestrator should be perform runtime
adaptation under certain condition. These models offer developers the option
to deploy and scale applications differently from the methods provided by
Kubernetes The outputs of the mentioned functions are stored in memory, giving
developers the flexibility to either save them as files or utilize them
directly.

###  II-B Termination Plans

To enable Converter to generate termination plans for a CEAML-described
application on Kubernetes or Kubevirt, developers must provide the following
required inputs:

  * •

The name of the running instance, for example acc-
uc2orbk-0-0-4-00036-gameserver-7reio-min1

  * •

The path of the model written in CEAML

Similarly to the deployment functions, all required inputs for the termination
function can be provided either manually or through a mechanism. A usage
example demonstrating how to generate the termination plans with Converter is
presented in Figure 4.

Figure 4: Code snippet for termination plans

As illustrated in the above code snippet, only the
Converter.undeploy(componentInfo, file_path) should be invoked. This method
internally calls the Parser, which parses the CEAML model and generates the
termination plan for the running instance.

###  II-C Scale Out Plans

To enable Converter to generate scale-out plans for a CEAML-described
application on Kubernetes or Kubevirt, an orchestrator or another mechanism
must retrieve the plans and deploy them with the same running instance ID to a
separate cluster. This scenario is exclusive to distributed orchestrators.
Developers should provide the following required inputs:

  * •

The name of the running instance, for example acc-
uc2orbk-0-0-4-00036-gameserver-7reio-min1

  * •

The path of the model written in CEAML

Similarly to the previously mentioned functions, all required inputs for the
scale-out function can be provided either manually or through a mechanism. A
usage example on how to generate the scale out plans to perform scaling with
Converter is presented in Figure 5.

Figure 5: Code snippet for scale out plans

The scale-out function, similar to the termination function, will trigger the
Parser in the background. The function will parse the CEAML model and generate
the scale-out plan for the running instance.

##  III Impact

Utilizing Converter enables platforms to leverage the descriptive capabilities
of CEAML language, facilitating deployment and runtime adaptation beyond the
functionalities offered by Kubernetes and Kubevirt alone. A distributed
orchestrator could employ the Converter in order to perform matchmaking
process and deployment of the described application components across
different Kubernetes or Kubevirt clusters.

CEAML and the Converter have already been used by a European project named
ACCORDION [6]. The distributed orchestrator [7] of the ACCORDION platform,
managed to carry out all orchestration processes supported by the Converter
across Kubernetes and Kubevirt clusters situated on both Edge and Cloud
environments. The orchestrator achieved a significant milestone by leveraging
CEAML and the Converter to autonomously execute deployment and runtime
adaptation, without depending on Kubernetes orchestrator and scaling
capabilities.

Following that, the Converter can assist in a range operations, spanning from
deploying an application to a single Kubernetes cluster to assisting
distributed orchestrators in deploying, terminating or scaling across multiple
Kubernetes and Kubevirt clusters.

##  IV Conclusion

Converter reasoner seamlessly integrates CEAML orchestration semantics into
both simple and complex orchestration mechanisms. Its primary advantage lies
in its ability to generate Kubernetes or Kubevirt configuration plans,
streamlining the orchestration process outlined in CEAML. Developers can
utilize the Converter’s outputs by applying them to a single Kubernetes
cluster or even to an orchestration mechanism overseeing multiple clusters.

The Converter can be easily installed as it is distributed as a Python
package. Its source code is openly available on a public GitHub repository,
which allows the cloud modeling languages community to delve into the code,
propose new features, or gain insight into the expansion of TOSCA to develop
CEAML.

## References

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