# On the cyclic coloring conjecture

Stanislav Jendrol<EMAIL_ADDRESS>Roman Soták<EMAIL_ADDRESS>Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice,
Slovakia

###### Abstract

A cyclic coloring of a plane graph $G$ is a coloring of its vertices such that
vertices incident with the same face have distinct colors. The minimum number
of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic
number $\chi_{c}(G)$. Let $\Delta^{*}(G)$ be the maximum face degree of a
graph $G$.

In this note we show that to prove the Cyclic Coloring Conjecture of Borodin
from 1984, saying that every connected plane graph $G$ has
$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor$, it is enough to do it
for subdivisions of simple $3$-connected plane graphs.

We have discovered four new different upper bounds on $\chi_{c}(G)$ for graphs
$G$ from this restricted family; three bounds of them are tight. As
corollaries, we have shown that the conjecture holds for subdivisions of plane
triangulations, simple $3$-connected plane quadrangulations, and simple
$3$-connected plane pentagulations with an even maximum face degree, for
regular subdivisions of simple $3$-connected plane graphs of maximum degree at
least 10, and for subdivisions of simple $3$-connected plane graphs having the
maximum face degree large enough in comparison with the number of vertices of
their longest paths consisting only of vertices of degree two.

###### keywords:

plane graphs, cyclic coloring, vertex-coloring

###### MSC:

[2010] 05C15 , 05C10

##  1 Introduction

Throughout this note we use graph theory terminology according to the books
[22] and [27]. However, we recall the most frequent notions. In this note, $G$
is a connected plane graph with vertex set $V(G)$, edge set $E(G)$, and face
set $F(G)$. In what follows, $G$ can have multiple edges but no loops, while a
simple graph has no multiple edges. The degree of a vertex $v$, denoted by
$\deg_{G}(v)$, is the number of edges incident with $v$. The degree of a face
$f$, denoted by $\deg_{G}(f)$, is the number of vertices incident with $f$. A
$k$-face is any face of degree $k$. We use $\Delta(G)$ and $\Delta^{*}(G)$ to
denote the maximum vertex degree and maximum face degree of $G$, respectively.
In a graph $G$, a subdivision of an edge $uv$ is the operation of replacing
$uv$ with a paths $u,w,v$ through a new vertex $w$. A subdivision of a graph
$G$ is a graph obtained by a sequence of subdivisions of edges of $G$.

For a cycle $C$ (in a plane graph $G$) we denote the set of vertices and edges
of $G$ lying inside $C$ and outside $C$ by $\mathrm{int}{{}_{G}(C)}$ and
$\mathrm{ext}{{}_{G}(C)}$, respectively. We say that $C$ is a separating cycle
if both $\mathrm{int}{{}_{G}(C)}$ and $\mathrm{ext}{{}_{G}(C)}$ are not empty.

Let $G$ be a connected plane graph and $C_{n}$ be a separating cycle of length
$n$ in $G$. Let $G_{1}$ (resp. $G_{2}$) be a plane graph obtained from $G$ by
deleting the exterior (resp. the interior) of $C_{n}$. Observe that
$\Delta^{*}(G_{i})\leq\max\\{\Delta^{*}(G),n\\}$, $i\in\\{1,2\\}$, where $n$
is the degree of the face $g_{i}$ of $G_{i}$ whose boundary is the cycle
$C_{n}$. The face $g_{i}$ is not a face of $G$ while any other face of $G$ is
present in $G_{1}$ or in $G_{2}$.

A cyclic coloring of a plane graph is a vertex coloring such that any two
different cyclically adjacent vertices, i.e. vertices incident with the same
face, receive distinct colors. The minimum number of colors needed for a
cyclic coloring of a graph $G$, the cyclic chromatic number, is denoted by
$\chi_{c}(G)$. This concept was introduced by Ore and Plummer [23].

It is obvious that any cyclic coloring of a connected plane graph $G$ requires
at least $\Delta^{*}(G)$ colors. As concerns the upper bound, Borodin [5] has
conjectured (see also Conjecture 6.1 in [7], or Section 2.5 in [19]):

######  Conjecture 1.

[5] If $G$ is a connected plane graph with maximum face degree
$\Delta^{*}(G)$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

As shown in [5], the upper bound in Conjecture 1 whenever true is best
possible.

This conjecture is known as the Cyclic Coloring Conjecture, see e.g. [14],
[15], or [20].

From the Four Color Theorem [3] and [4] it follows that $\chi_{c}(G)\leq 4$ if
$\Delta^{*}(G)=3$. From Borodin’s proof of the Ringel’s conjecture [5] (see
also [6]) we have $\chi_{c}(G)\leq 6$ if $\Delta^{*}(G)\leq 4$. Both these
bounds are tight. Hebdige and Král’ proved this conjecture for
$\Delta^{*}(G)=6$, see [16]. So in this note we shall study only the cases
when $\Delta^{*}(G)\geq 5$. Ore and Plummer [23] proved that $\chi_{c}(G)\leq
2\Delta^{*}(G)$, which was improved to
$\lfloor\frac{9}{5}\Delta^{*}(G)\rfloor$ by Borodin, Sanders, and Zhao [9],
and to $\lceil\frac{5}{3}\Delta^{*}(G)\rceil$ by Sanders and Zhao [25].
Moreover, Borodin at al. [9] proved that $\chi_{c}(G)\leq 8$ for
$\Delta^{*}(G)=5$. Amini, Esperet, and van den Heuvel [1] proved that the
conjecture holds asymptotically in the following sense: for every
$\epsilon>0$, there exists $\Delta_{\epsilon}$ such that every plane graph
with maximum face degree $\Delta^{*}\geq\Delta_{\epsilon}$ has a cyclic
coloring with at most $(\frac{3}{2}+\epsilon)\Delta^{*}$ colors.

A beautiful result was obtained by Borodin, Broersma, Glebov, and van den
Heuvel [8] using the following parameter $k^{*}=k^{*}(G)$, which denotes the
maximum number of vertices that two faces of $G$ can have in common.

######  Theorem 1.

[8] For every connected plane graph $G$ with $\Delta^{*}(G)\geq 5$ it holds

$\chi_{c}(G)\leq\max\\{\Delta^{*}(G)+3k^{*}+2,\Delta^{*}(G)+14\\}.$

They also posed the following:

######  Conjecture 2.

[8] Every plane graph $G$ with $\Delta^{*}(G)$ and $k^{*}$ large enough has a
cyclic coloring with

$\Delta^{*}(G)+k^{*}$

colors.

Better bounds are known for simple $3$-connected plane graphs. The first bound
for such graphs, $\chi_{c}(G)\leq\Delta^{*}(G)+9$, was obtained by Plummer and
Toft [24]. They also conjectured that any simple $3$-connected plane graph has
$\chi_{c}(G)\leq\Delta^{*}(G)+2$. The best presently known upper bounds for
the cyclic chromatic number of simple $3$-connected plane graphs are:
$\chi_{c}(G)\leq\Delta^{*}(G)+1$ for $\Delta^{*}(G)\geq 122$ by Borodin and
Woodall [10] and for $\Delta^{*}(G)\geq 60$ by Enomoto, Horňák, and Jendrol’
[13], $\chi_{c}(G)\leq\Delta^{*}(G)+2$ for $\Delta^{*}(G)\geq 24$ by Horňák
and Jendrol’ [17], for $\Delta^{*}(G)\geq 18$ by Horňák and Zlámalová [18],
for $\Delta^{*}(G)\geq 16$ by Dvořák, Hebdige, Hlásek, Král’, and Noel [11],
for $\delta(G)=4$ and $\Delta^{*}(G)\geq 6$, or $\delta(G)=5$ by Zlámalová
[28], and for $G$ being locally connected by Kriesell [21]. Azarija, Erman,
Král’, Krnc, and Stacho [2] proved that for every plane graph $G$, in which
any two faces of degree at least four are vertex disjoint, holds
$\chi_{c}(G)\leq\Delta^{*}(G)+1$. Havet, Sereni, and Škrekovski [15] proved
that $\chi_{c}(G)\leq 11$ for $\Delta^{*}(G)=7$. The best presently known
general upper bound, $\chi_{c}(G)\leq\Delta^{*}(G)+5$, is by Enomoto and
Horňák [12]. Summarizing we have:

######  Theorem 2.

If $G$ is a simple $3$-connected plane graph, then
$\chi_{c}(G)\leq\Delta^{*}(G)+r$, where

  1. 1.

$r=1$ if $\Delta^{*}(G)\geq 60$, or $\Delta^{*}(G)=3$, or any two faces of $G$
of degree at least four are vertex disjoint,

  2. 2.

$r=2$ if $\Delta^{*}(G)\geq 16$, or $\Delta^{*}(G)=4$, or $\delta(G)=4$ and
$\Delta^{*}(G)\geq 6$, or $\delta(G)=5$, or $G$ is locally connected,

  3. 3.

$r=3$ if $5\leq\Delta^{*}(G)\leq 6$,

  4. 4.

$r=4$ if $\Delta^{*}(G)=7$, and

  5. 5.

$r=5$ in the remaining cases.

In this paper we show that to prove Conjecture 1 it is enough to do it for
subdivisions of simple $3$-connected plane graphs. Next we have obtained four
different upper bounds on $\chi_{c}(G)$ for graphs from this family; three of
them are tight. As corollaries we show that Conjecture 1 holds for large
maximum face degree subdivisions of simple $3$-connected plane graphs, and for
four wide families of plane graphs without restrictions on maximum face
degrees.

##  2 Definitions and some remarks about the structure of plane graphs

Throughout this paper $P_{m}$ will denote a path of length $m$. By $t=t(G)$ we
will denote the number of vertices of a longest path in $G$ all vertices of
which have degree 2.

For a $2$-connected plane graph $G$ other than cycle the reduction $R(G)$ of
$G$ is a graph obtained from $G$ by replacing all maximal $u,v$-paths whose
all interior vertices are of degree 2 with the edges $uv$. More precisely,
$V(R(G))=\\{v\in V(G)|\deg_{G}(v)\geq 3\\}=V_{\geq 3}(G)$ and $E(R(G))=P(G)$
which is the set of all maximal paths of $G$ all interior vertices of which
have degree 2 and whose ends are vertices of degree at least 3.

Observe, that $|F(G)|=|F(R(G))|$ and that there is a one-to-one correspondence
between sets of faces of $F(G)$ and $F(R(G))$, between the set of edges
$E(R(G))$ and the set of paths $P(G)$, and between the sets of vertices
$V(R(G))$ and $V_{\geq 3}(G)$. Clearly, $R(G)$ is a $2$-connected plane graph
with minimum degree 3.

Observe, that $R(G)$ has at least one of the following properties:

  1. 1.

It is a simple $3$-connected graph.

  2. 2.

It contains a 2-face [u,v].

  3. 3.

It has a vertex-cut $\\{u,v\\}$ and a component $K$ of
$R(G)\setminus\\{u,v\\}$ such that a $2$-connected subgraph $H_{R}(u,v)$ on
the vertex set $V(K)\cup\\{u,v\\}$ has the following structure: All vertices
and edges of $H_{R}(u,v)$ lie on or in the interior (resp. in the exterior) of
a separating cycle $C^{\prime}$ determined by two internally vertex disjoint
$u,v$-paths $P^{\prime}$ and $P^{\prime\prime}$ whose all internal (resp.
external) vertices are from the set $V(K)$.

Note, that it is enough to choose a vertex-cut $\\{u,v\\}$ of minimal number
of vertices of $V(K)$. Then $H_{R}(u,v)$ is the subgraph of $R$ consisting of
the vertex set $V(K)\cup\\{u,v\\}$ and the edge set containing $E(K)$ and all
edges of $R$ between the vertices of $V(K)$ and the vertices of $\\{u,v\\}$
(and no edge $uv$). From the minimality of $K$ it follows that $H_{R}(u,v)$ is
$2$-connected. The boundary cycle of $H_{R}(u,v)$ containing both $u$ and $v$
is chosen as a desired cycle $C^{\prime}$.

##  3 A structural Theorem

######  Theorem 3.

If $G$ is a $2$-connected plane graph with maximum face degree $\Delta^{*}(G)$
and at least four faces, then $G$ is either a subdivision of a simple
$3$-connected plane graph or contains a separating cycle $C_{n}$ with
$n\leq\Delta^{*}(G)$.

###### Proof.

Let $\Delta^{*}=\Delta^{*}(G)$. It is easy to see that the theorem holds if
$G$ has exactly four faces or it contains a $2$-face. Next we distinguish two
cases depending on the structure of the reduction $R=R(G)$.

If $G$ is a subdivision of a simple $3$-connected plane graph (i.e. $R$ is a
simple $3$-connected plane graph), then there is nothing to prove.

Otherwise the reduction $R$ contains either a $2$-face [u,v] or has a suitable
vertex-cut {u, v}. Consider $H_{G}(u,v)$, the subgraph of $G$ corresponding to
the $2$-face [u,v] or to the subgraph $H_{R}(u,v)$ of $R$.

Let the cycle $C_{a+b}$, bounding $H_{G}(u,v)$ in $G$, be defined by the
$u,v$-paths $P_{a}$ and $P_{b}$ corresponding to the edges $uv$ of the
$2$-face $[u,v]$ or to the $u,v$-paths $P^{\prime}$ and $P^{\prime\prime}$ of
the subgraph $H_{R}(u,v)$, respectively, of $R$. Let $f_{1}$ and $f_{2}$ be
the faces of $G$ in the exterior of $C_{a+b}$ defined by the paths $P_{a}$ and
$P_{b}$, respectively. Let $P_{c}$ and $P_{d}$ be the other $u,v$-paths
bounded the faces $f_{1}$ and $f_{2}$, respectively. This is always possible
because in every $2$-connected plane graph any face is bounded by a cycle.
Observe that $\deg_{G}(f_{1})=a+c\leq\Delta^{*}$ and
$\deg_{G}(f_{2})=b+d\leq\Delta^{*}$. Let, w.l.o.g., $b\geq a$. Then
$a+d\leq\Delta^{*}$ and there is, in $G$, a separating cycle $C_{n}$ of length
at most $\Delta^{*}$, namely $C_{n}=C_{a+d}$.

∎

##  4 Two Lemmas

######  Lemma 1.

If $G$ is a $2$-connected plane graph with exactly three faces and maximum
face degree $\Delta^{*}(G)$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

###### Proof.

The graph $G$ consists of three edge disjoint paths $P_{a}$, $P_{b}$, and
$P_{c}$ joining two vertices $u$ and $v$. Because for any two vertices there
is, in $G$, a common face incident with both of them, we have
$\chi_{c}(G)=a+b+c-1$. Observe that $a+b\leq\Delta^{*}(G)$,
$a+c\leq\Delta^{*}(G)$, and $c+b\leq\Delta^{*}(G)$. This implies

$2(a+b+c-1)+2=2\chi_{c}(G)+2\leq 3\Delta^{*}(G)$

which immediately gives the statement of the lemma.

∎

The construction of the following lemma first appears in [5].

######  Lemma 2.

For every $t\geq 0$ there exists a $2$-connected plane graph $H$ of maximum
face degree $\Delta^{*}(H)$ with the reduction $R(H)$ being simple
$3$-connected plane graph, $t=t(H)$, and

$\chi_{c}(H)=\Delta^{*}(H)+t+2=\lfloor\frac{3}{2}\Delta^{*}(H)\rfloor.$

###### Proof.

The triangular prism $D_{3}$ with three edges joining the two triangles
replaced by disjoint paths of equal length $t+1,t\geq 0$, gives a graph $H$
with $\chi_{c}(H)=\Delta^{*}(H)+t+2=\lfloor\frac{3}{2}\Delta^{*}(H)\rfloor.$ ∎

##  5 Reduction of Conjecture 1

To prove Conjecture 1 it is enough to prove the following:

######  Conjecture 3.

If $G$ is subdivision of a simple $3$-connected plane graph with maximum face
degree $\Delta^{*}(G)$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

######  Theorem 4.

Conjecture 1 holds if and only if Conjecture 3 holds.

###### Proof.

It is enough to prove that Conjecture 1 follows from Conjecture 3. Suppose it
is not true.

Let $H$ be a counterexample with minimum number of faces and then with the
minimum number of vertices. It is easy to see that $H$ is $2$-connected, has
at least four faces (because Lemma 1), and is not a subdivision of any simple
$3$-connected plane graph. Then, by Theorem 3, $H$ contains a separating cycle
$C_{n}$, with $n\leq\Delta^{*}(H)$. The graphs
$H_{1}=C_{n}\cup\mathrm{int}{{}_{G}(C_{n})}$ and
$H_{2}=C_{n}\cup\mathrm{ext}{{}_{G}(C_{n})}$, are smaller than $H$, therefore,
for $i\in\\{1,2\\}$ we have

$\chi_{c}(H_{i})\leq\lfloor\frac{3}{2}\Delta^{*}(H_{i})\rfloor.$

Since the graphs $H_{1}$ and $H_{2}$ have only vertices of the cycle $C_{n}$
in common, $n\leq\Delta^{*}(H)$, and each face of $H$ is also a face in
$H_{1}$ or in $H_{2}$, there is $\Delta^{*}(H_{i})\leq\Delta^{*}(H)$. So we
can combine the cyclic colorings of $H_{1}$ and $H_{2}$ to obtain a cyclic
coloring of $H$ using at most $\frac{3}{2}\Delta^{*}(H)$ colors. A
contradiction. ∎

##  6 Cyclic colorings of subdivisions of simple $3$-connected plane graphs

As in any simple $3$-connected plane graph every two distinct faces have at
most one edge in common, for any subdivision $G$ of a simple $3$-connected
plane graph we have $k^{*}=t+2,t=t(G)$. As a result, from Theorem 1, we have:

######  Theorem 5.

If $G$ is a subdivision of a simple $3$-connected plane graph with
$\Delta^{*}(G)\geq 5$, then

$\chi_{c}(G)\leq\max\\{\Delta^{*}(G)+3t+8,\Delta^{*}(G)+14\\}.$

######  Corollary 1.

If $G$ is a subdivision of a simple $3$-connected plane graph with
$\Delta^{*}(G)\geq\max\\{6t+16,28\\}$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

Let $G$ be a $2$-connected plane graph with the reduction $R=R(G)$ being a
simple $3$-connected plane graph. We associate with the graph $G$ the
following plane multigraph $S=S(G)$, called the subdivision multigraph of $G$,
whose vertex set $V(S)=F(G)$, the face set of $G$, and edge set
$E(S)=V_{2}(G)$, the set of vertices of degree 2 of $G$. The edge $v$ joins
the vertices $f_{a}$ and $f_{b}$ in $S$ if and only if the 2-vertex $v$ is
incident, in $G$, with both faces $f_{a}$ and $f_{b}$. Observe that the
multigraph $S$ has the maximum vertex degree $\Delta(S)=\max_{f\in
F(G)}\\{\deg_{G}(f)-\deg_{R}(f^{\prime})\\}$ where $f^{\prime}$ is the face of
$R$ corresponding to the face $f$ of $G$. Let $\chi^{\prime}(S)$ denote the
chromatic index of the subdivision multigraph of $S$.

######  Theorem 6.

Let $G$ be a subdivision of a simple $3$-connected plane graph $R$ (which is
the reduction $R=R(G)$) and let $S=S(G)$ be the subdivision multigraph of $G$.
Then

$\chi_{c}(G)\leq\chi^{\prime}(S)+\chi_{c}(R).$

Moreover, the bound is tight.

###### Proof.

Let $\phi:V(R)\rightarrow\\{1,\dots,\chi_{c}(R)\\}$ be a cyclic coloring of
$R$.

Let
$\psi:V_{2}(G)=E(S)\rightarrow\\{\chi_{c}(R)+1,\dots,\chi_{c}(R)+\chi^{\prime}(S)\\}$
be a proper edge-coloring of $S$ with $\chi^{\prime}(S)$ colors. The edge-
coloring $\psi$ induces a vertex-coloring of vertices of $V_{2}(G)$ in which
every two distinct vertices $v_{1}$ and $v_{2}$ of degree 2, that are incident
with the same face $f$, receive different colors. It is easy to see that the
colorings $\phi$ and $\psi$ together give a cyclic coloring of $G$.

The bound is tight for any $2$-connected plane graph in which any two its
vertices are cyclically adjacent. For an example of such a graph $H$ consider
the triangular prism $D_{3}$ with three edges joining two triangles replaced
by three paths of lengths $a+1$, $b+1$, and $c+1$. It is easy to see that

$\chi_{c}(H)=(a+b+c)+6=\chi^{\prime}(S(H))+\chi_{c}(R(H)).$

∎

######  Theorem 7.

If $G$ is a subdivision of a $3$-connected plane graph $R$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\max_{f\in
F(G)}\\{\deg_{G}(f)-\deg_{R}(f^{\prime})\\}\rfloor+\chi_{c}(R),$

where $f^{\prime}$ is the face of $R$ corresponding to the face $f$ of $G$.
Moreover, the bound is tight.

###### Proof.

By theorem of Shannon [26] it holds that
$\chi^{\prime}(S)\leq\lfloor\frac{3}{2}\Delta(S)\rfloor.$ To see the
tightness, consider the graph $H$ of the proof of Theorem 6 with $a=b=c\geq
2$. ∎

Next corollary provides three other wide families of graphs for which
Borodin’s conjecture holds.

######  Corollary 2.

If $G$ is a subdivision of a plane triangulation, a simple $3$-connected plane
quadrangulation, or a simple $3$-connected plane pentagulation (all faces are
of degree 5) with even maximum face degree, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

###### Proof.

We apply Theorem 7. If $G$ is a triangulation, then
$\chi_{c}(G)\leq\lfloor\frac{3}{2}(\Delta^{*}(G)-3)\rfloor+4=\lfloor\frac{3\Delta^{*}-9+8}{2}\rfloor\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

If $G$ is a simple $3$-connected plane quadrangulation, then
$\chi_{c}(G)\leq\lfloor\frac{3}{2}(\Delta^{*}(G)-4)\rfloor+6=\lfloor\frac{3\Delta^{*}-12+12}{2}\rfloor=\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

If $G$ is a subdivision of a simple $3$-connected plane pentagulation of even
maximum face degree, then
$\chi_{c}(G)\leq\lfloor\frac{3}{2}(\Delta^{*}(G)-5)\rfloor+8=\lfloor\frac{3\Delta^{*}-15+16}{2}\rfloor=\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

∎

##  7 One more upper bound on cyclic chromatic number

######  Theorem 8.

If $G$ is a subdivision of a simple $3$-connected plane graph $R$, then

$\chi_{c}(G)\leq\max_{f\in
F(G)}\\{\deg_{G}(f)-\deg_{R}(f^{\prime})\\}+t(G)+\chi_{c}(R),$

where $f^{\prime}$ is the face of $R$ corresponding to the face $f$ of $G$.
Moreover, the bound is tight.

###### Proof.

We use for $\chi^{\prime}(S)$ in the proof of Theorem 7, instead of the
Shannon bound, the bound of Vizing and Gupta (see [27], p. 275), saying that
$\chi^{\prime}(S)\leq\Delta(S)+\mu(S)$ where $\mu(S)$ denotes the maximum edge
multiplicity of $S$, and the fact that $\mu(S)=t(G)$. To see the tightness,
consider the graph $H$ from the proof of Theorem 6 with $a=b=c=t(H)\geq 2$. ∎

######  Corollary 3.

If $G$ is a subdivision of a simple $3$-connected plane graph $R$ with
$\Delta^{*}(G)\geq 2\chi_{c}(R)+2t(G)-6$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

Note that if $\Delta^{*}(R)\leq 2t(G)+6$ for $t\geq 2$ or $\Delta^{*}(R)\leq
12-t$ for $t\leq 1$, then the bound of Theorem 8 is better than that of
Theorem 5 (and, hence, of Theorem 1).

The graph $G$ is a regular subdivision of a graph $R$ if it is obtained from
$R$ by replacing each edge of $R$ with a path of length $k+1$ for some
constant $k\geq 0$.

######  Corollary 4.

If $G$ is a regular subdivision of a simple $3$-connected plane graph $R$,
then

$\chi_{c}(G)\leq\Delta^{*}(G)+t(G)+r,$

where $r$ is a constant depending on the properties of $R$, see Theorem 2.

###### Proof.

It is easy to see that for any face $f$ of $G$ there is
$\deg_{G}(f)=\deg_{R}(f^{\prime})(k+1)$ for some integer $k\geq 0$, where
$f^{\prime}$ is the face of $R$ corresponding to the face $f$. Then $t(G)=k$
and, by Theorem 8, $\chi_{c}(G)\leq\max_{f\in
F(G)}\\{\deg_{G}(f)-\deg_{R}(f^{\prime})\\}+t(G)+\Delta^{*}(R)+r=\max_{f\in
F(G)}\\{k\deg_{R}(f^{\prime})\\}+t(G)+\Delta^{*}(R)+r=\Delta^{*}(R)(k+1)+t(G)+r=\Delta^{*}(G)+t(G)+r.$

∎

######  Corollary 5.

If $G$ with $t(G)\geq 1$ is a regular subdivision of a simple $3$-connected
plane graph $R$, then

$\chi_{c}(G)\leq\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

###### Proof.

Let $G$ be a regular subdivision of the graph $R$ with $t(G)=k\geq 1$. From
$\Delta^{*}(G)\geq 2+\frac{8}{k+1}$ and Corollary 4, we have
$\chi_{c}(G)\leq\Delta^{*}(R)(k+1)+k+5\leq\lfloor\frac{3}{2}\Delta^{*}(R)(k+1)\rfloor=\lfloor\frac{3}{2}\Delta^{*}(G)\rfloor.$

∎

##  8 One new conjecture

We believe that the following conjecture, which involves both, the Plummer and
Toft conjecture and the corresponding part of Conjecture 2, holds:

######  Conjecture 4.

If $G$ is a subdivision of a simple $3$-connected plane graph, then

$\chi_{c}(G)\leq\Delta^{*}(G)+t(G)+2.$

Observe, that Conjecture 4 holds for regular subdivisions of simple
$3$-connected plane graphs $R$ with $\Delta^{*}(R)\geq 16$, or
$\Delta^{*}(R)\leq 4$, or $\delta(R)=4$ and $\Delta^{*}(R)\geq 6$, or
$\delta(R)=5$, or $R$ being locally connected (see Corollary 4 and Theorem 2,
cases $r=1$ and $r=2$), for subdivisions of plane triangulations, and for
subdivisions of simple $3$-connected plane quadrangulations.

In the case of triangulations we have from Theorem 8:

$\chi_{c}(G)\leq\Delta^{*}(G)-3+t(G)+3+1=\Delta^{*}(G)+t(G)+1.$

In the case of quadrangulations Theorem 8 provides:

$\chi_{c}(G)\leq\Delta^{*}(G)-4+t(G)+4+2=\Delta^{*}(G)+t(G)+2.$

From Lemma 2 we know that the upper bound on $\chi_{c}(G)$ in our Conjecture 4
cannot be improved.

Acknowledgments

This work was supported by the Slovak Research and Development Agency under
the Contract No. APVV-19-0153.

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# A comparison between Avila-Gouëzel-Yoccoz norm and Teichmüller norm

Weixu Su  and  Shenxing Zhang  Weixu Su: School of Mathematics, Sun Yat-sen
University, Guangzhou 510275, China<EMAIL_ADDRESS>Shenxing, Zhang:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>
###### Abstract.

We give a comparison between the Avila-Gouëzel-Yoccoz norm and the Teichmüller
norm on the principal stratum of holomorphic quadratic differentials.

##  1\. Introduction

Let $X$ be a compact Riemann surface of genus $g$. A holomorphic quadratic
differential $q$ on $X$ is a tensor given locally by an expression
$q=q(z)dz^{2}$ where $z$ is a conformal coordinate on $X$ and $q(z)$ is
holomorphic. Such a (nonzero) quadratic differential $q$ defines a flat metric
$|q|^{1/2}$ on $X$. This metric has conical singularities at zeroes of $q$.
Its area is defined by

$\|q\|=\int_{X}|q(z)||dz|^{2}.$

Fix $g\geq 2$ and let $\mathcal{P}_{g}$ be the _principal stratum_ of the
moduli space of quadratic differentials, consisting of isomorphism classes of
holomorphic quadratic differentials $(X,q)$ with $4g-4$ distinct simple
zeroes.

There is a Finsler metric on $\mathcal{P}_{g}$ called _AGY metric_ , which was
introduced by Avila-Gouëzel-Yoccoz [2, §2.2.2] on each stratum of Abelian
differentials. This norm plays an important role in the study of Teichmüller
flow. See [1, 2, 4].

Let $\mathcal{M}_{g}$ be the moduli space of Riemann surfaces of genus $g$.
Let $\pi:\mathcal{P}_{g}\to\mathcal{M}_{g}$ be the natural projection, defined
by $\pi(X,q)=X$. In the note, we consider the derivative of $\pi$ and compare
the AGY norm with the Teichmüller norm.

For each $(X,q)\in\mathcal{P}_{g}$, there is a _canonical double cover_
$\rho:\hat{X}\to X$, ramified at the odd zeros of $q$, such that $\rho^{*}q$
is the square of an Abelian differential $\omega$ on $\hat{X}$. See [3] or [9,
§2] for details. The Abelian differential $\omega$ is a $-1$ eigenvector for
the holomorphic involution $\tau:\hat{X}\to\hat{X}$ that permutes the sheets
of the double cover, that is,

$\rho^{*}\omega=-\omega.$

We can identify the tangent space of $\mathcal{P}_{g}$ at $(X,q)$ as
$H^{1}_{-1}(\hat{X},\mathbb{C})$, the $-1$ eigenspace for the action of $\tau$
on the cohomology $H^{1}(\hat{X},\mathbb{C})$.

Every element of $H^{1}(\hat{X},\mathbb{C})$ can be represented uniquely by a
harmonic one-form. Consequently, there is a natural decomposition of
$H^{1}_{-1}(\hat{X},\mathbb{C})$ into $H_{-1}^{1,0}(\hat{X})\oplus
H_{-1}^{0,1}(\hat{X}).$ Note that the kernel of $D\pi$ is
$H_{-1}^{1,0}(\hat{X})$. See Theorem 2.1 below.

We consider $\eta\in H_{-1}^{0,1}(\hat{X})$ and compare the AGY norm of $\eta$
with the Teichmüller norm of $D\pi(\eta)$. The main result is:

######  Theorem 1.1.

Let $(X,q)\in\mathcal{P}_{g}$ with area $\|q\|=1$. Let $\rho:\hat{X}\to X$ be
the canonical double cover such that $\rho^{*}q=\omega^{2}$. Then for any
$\eta\in H_{-1}^{0,1}(\hat{X})$, we have

(1) $\frac{r}{\sqrt{2}}\
{\|\eta\|_{\mathrm{AGY}}}\leq{\|D\pi(\eta)\|_{\mathrm{Teich}}}\leq\frac{8}{\sqrt{\pi}r}\|\eta\|_{\mathrm{AGY}},$

where $2r$ is the shortest length of saddle connections on $(\hat{X},\omega)$.

######  Remark 1.2.

Note that the area of $\omega$ is $2$.

Recently, Kahn-Wright [6] derived a comparison between the Hodge norm (another
important norm on $\mathcal{P}_{g}$) and the Teichmüller norm. Our research is
motivated by their work.

The paper has the following structure. In §2, we present some basic properties
of quadratic differentials. The upper bound in (1) is given in §3, where we
use the Delaunay triangulation of quadratic differential to construct
quasiconformal maps with explicit Beltrami differentials. In §4, we give an
upper bound of the AGY norm in terms of the Hodge norm, and then we derive the
lower bound in (1) from Kahn-Wright[6, Theorem 1.4].

##  2\. Preliminaries

###  2.1. The moduli space of quadratic differentials

Let $g\geq 2$. We denote by $\mathcal{M}_{g}$ the moduli space of compact
Riemann surfaces of genus $g$. For $X\in\mathcal{M}_{g}$, the cotangent space
of $\mathcal{M}_{g}$ at $X$ is canonically identified with the space $Q(X)$ of
holomorphic quadratic differentials on $X$. We define the $L^{1}$-norm on
$Q(X)$ by

$\|q\|=\int_{X}|q|.$

A tangent vector of $\mathcal{M}_{g}$ at $X$ is represented by a Beltrami
differential $\mu$. There is a natural pairing between quadratic differentials
and Beltrami differentials given by

$\langle\mu,q\rangle=\int_{X}\mu q.$

The _Teichmüller norm_ of $\mu$ is defined by

$\|\mu\|_{\mathrm{Teich}}=\sup_{\|q\|=1}\operatorname{Re}\
\langle\mu,q\rangle.$

This gives the infinitesimal form of the Teichmüller metric on
$\mathcal{M}_{g}$.

Let $\mathcal{Q}_{g}$ be the moduli space of quadratic differentials,
consisting of pairs $(X,q)$ where $X$ is a compact Riemann surface of genus
$g$ and $q$ is a holomorphic quadratic differential on $X$. The moduli space
$\mathcal{Q}_{g}$ has a stratified structure: given an integral vector
$\kappa=\left(\kappa_{1},\cdots,\kappa_{n}\right)$ with $\sum\kappa_{i}=4g-4$,
we let $\mathcal{Q}_{g}(\kappa)\subset\mathcal{Q}_{g}$ be the set of quadratic
differentials $(X,q)$ where $q$ has $n$ zeros of order
$\kappa_{1},\cdots,\kappa_{n}$.

In the paper, our study is mainly restricted on the principal stratum,
consisting of those quadratic differentials all of whose zeros are simple. We
denote the principal stratum by $\mathcal{P}_{g}$. This stratum is both open
and dense in $\mathcal{Q}_{g}$.

###  2.2. Canonical double cover

Let $\mathcal{Q}_{g}(\kappa)$ be a stratum of quadratic differentials. Given
$(X,q)\in\mathcal{Q}_{g}(\kappa)$, let $\rho:\hat{X}\to X$ be the canonical
double cover such that the pull-back $\rho^{*}q$ becomes the square of an
Abelian differential $\omega$ on $\hat{X}$. Let $\tau:\hat{X}\to\hat{X}$ be
the involution that permutes the sheets of the double cover. By the
construction, $\tau^{*}\omega=-\omega$.

Let $\Sigma$ be the set of zeros of $\omega$. Denote by
$H^{1}_{-1}(\hat{X},\Sigma,\mathbb{C})$ the $-1$ eigenspace for the action of
$\tau$ on the relative homology group $H^{1}(\hat{X},\Sigma,\mathbb{C})$. Note
that the relative cohomology class of $\omega$ is an element of
$H^{1}_{-1}(\hat{X},\Sigma,\mathbb{C})$. A neighborhood of $\omega$ in
$H^{1}_{-1}(\hat{X},\Sigma,\mathbb{C})$ gives a local chart of $q$ in the
stratum, via the period mapping.

In the following, we shall identify the tangent space at $(X,q)$ with the
cohomology $H^{1}_{-1}(\hat{X},\Sigma,\mathbb{C})$. If
$(X,q)\in\mathcal{P}_{g}$, then $q$ has no zeros of even order. In this case,
since $\Sigma$ is the set of fixed points of $\tau$, we have

$H^{1}_{-1}(\hat{X},\Sigma,\mathbb{C})\cong H^{1}_{-1}(\hat{X},\mathbb{C}).$

Thus each element of $H^{1}_{-1}(\hat{X},\mathbb{C})$ can be uniquely
represented by a harmonic $1$-form.

The following result describe the tangent map of
$\pi:\mathcal{P}_{g}\to\mathcal{M}_{g}$ in terms of the period coordinates. It
is proved by Kahn-Wright [6, Corollary 1.2].

######  Theorem 2.1.

Consider the projection $\pi:\mathcal{P}_{g}\to\mathcal{M}_{g}$. Let
$(X,q)\in\mathcal{P}_{g}$ and let $\eta$ be a harmonic $1$-form on $\hat{X}$
that represents an element of $H^{1}_{-1}(\hat{X},\mathbb{C})$. Then for any
$\phi\in Q(X)$,

$\langle
D\pi(\eta),\phi\rangle=\frac{1}{2}\int_{\hat{X}}\rho^{*}(\phi)\frac{\eta^{0,1}}{\omega},$

where $\eta^{0,1}$ is the anti-holomorphic part of $\eta$.

###  2.3. The AGY norm.

The AGY norm is defined by Avila-Gouezel-Yoccoz [2] on any stratum of Abelian
differentials.

With the notations in §2.2, we consider the Abelian differential $\omega$ as
an element of $H^{1}(\hat{X},\Sigma,\mathbb{C})$. A saddle connection of
$\omega$ is a geodesic segment for the flat metric defined by $|\omega|$
joining two zeros of $\omega$ and not passing any zero in its interior. Each
saddle connection $\gamma$ gives rise to an element $[\gamma]$ of
$H^{1}(\hat{X},\Sigma,\mathbb{C})$. Denote by $\left\\{\gamma_{j}\right\\}$
the set of saddle connections on $\omega$. Then for any $[\eta]\in
H^{1}(\hat{X},\Sigma,\mathbb{C})$, its AGY norm is defined by

$\|\eta\|_{\mathrm{AGY}}=\sup\frac{\left|\int_{\gamma_{j}}\eta\right|}{\left|\int_{\gamma_{j}}\omega\right|},$

where the supremum is taken over all saddle connections.

Avila-Gouezel-Yoccoz [2] showed that the AGY norm is continuous and induces a
complete metric on the stratum.

##  3\. The upper bound

In this section, we give an upper bound of $\|D\pi(\eta)\|_{\mathrm{Teich}}$
in terms of $\|\eta\|_{\mathrm{AGY}},$ for any $\eta\in
H^{1}_{-1}(\hat{X},\mathbb{C})$. The idea is to triangulate the surface and
compute the Beltrami differentials of maps that are affine on each triangle.
We remark that the proof applies to any other stratum of quadratic
differentials or Abelian differentials.

###  3.1. Delaunay triangulation

Given a quadratic differential $(X,q)$, there is an associated flat metric
(with conical singularities) on $X$, defined by $|q|^{1/2}$. Denote by
$\Sigma$ the set of zeros of $q$. For any $x\in X$, let $d(x,\Sigma)$ be the
minimal $|q|^{1/2}$-distance from $x$ to $\Sigma$.

The next result is proved by Masur-Smillie [8]. See also Farb-Masur [5].

######  Proposition 3.1.

Let $(X,q)$ be a holomorphic quadratic differential of area $\|q\|\leq 1$.
There is a triangulation $\Delta$ on $X$ with the following properties:

  1. (1)

The vertices of $\Delta$ lie in the zero set of $q$.

  2. (2)

The edges of $\Delta$ are saddle connections of $q$.

  3. (3)

Each triangle is inscribed in a circle of radius $d(x,\Sigma)$ for some $x\in
X$.

The above construction is called a Delaunay triangulation of $q$.

Let $s=\sqrt{\frac{2}{\pi}}$, and let $B_{s}$ be the set of points in $X$ with
$d(x,\Sigma)\leq s$. By the proof of [8, Theorem 5.3], the complement of
$B_{s}$ is contained in a union of disjoint maximal flat cylinders, with the
property that their circumference is less than their height.

###  3.2. The proof of upper bound

Let $\eta\in H^{1}_{-1}(\hat{X},\mathbb{C})$. Denote by
$(\hat{X}_{t},\omega_{t})$ the family of Abelian differentials corresponding
to the cohomology classes $\omega+t\eta\in H^{1}_{-1}(\hat{X},\mathbb{C})$,
for sufficiently small $t>0$.

Let $\Delta$ be a Delaunay triangulation of $(\hat{X},\omega)$. By the
construction, the vertices of $\Delta$ are the zeros of $\omega$, and the
edges of $\Delta$ are saddle connections of $\omega$. For each $t$, we can
straighten $\Delta$ to be a triangulation of $\hat{X}_{t}$, denoted by
$\Delta_{t}$, such that the edges are saddle connections of $\omega_{t}$.

The next step is to construct quasiconformal mappings $f_{t}$ from $\hat{X}$
to $\hat{X}_{t}$ that are linear on each triangle. Denote the Beltrami
differentials of $f_{t}$ by $\mu_{t}$. Then

$D\pi(\eta)\cong\frac{d\mu_{t}}{dt}|_{t=0}.$

######  Proposition 3.2.

Let $2r$ be the shortest length of saddle connections on $(\hat{X},\omega)$.
Then

$\left\|D\pi(\eta)\right\|_{\mathrm{Teich}}\leq\frac{8}{\sqrt{\pi}r}\|\eta\|_{\mathrm{AGY}}.$

###### Proof.

Denote by

$\mu=\frac{d\mu_{t}}{dt}|_{t=0}.$

Since $\|\mu\|_{\mathrm{Teich}}\leq\|\mu\|_{\infty}$, it suffices to give the
upper bound for $\|\mu\|_{\infty}$.

Let $T=\triangle OAB$ be any triangle of $\Delta$, where $O,A,B$ denotes the
vertices. For simplicity, we consider $T$ as a triangle in the complex plane
and put $O=0,A=a>0$ and $B=b\in\mathbb{C}$. By definition,

$a=\int_{\gamma}\omega,b=\int_{\gamma^{\prime}}\omega,$

where $\gamma$ and $\gamma^{\prime}$ denote the saddle connection connecting
$O$ to $A$ and $O$ to $B$, respectively.

For each $t$ sufficiently small, the corresponding triangle in $\hat{X}_{t}$
has vertices given by $0$, $a+t\alpha$ and $b+t\beta$, where

$\alpha=\int_{\gamma_{i}}\eta,\beta=\int_{\gamma_{j}}\eta.$

Denote the associated affine mapping between the triangles by

$f_{t}(z)=Rz+S\bar{z}.$

Then we have

$Ra+Sa=a+t\alpha,$ $Rb+S\bar{b}=b+t\beta.$

A simple computation shows that the Beltrami coefficient $\mu_{t}$ is equal to

$\frac{S}{R}=t\frac{\frac{\alpha}{a}-\frac{\beta}{b}}{1-\frac{\bar{b}}{b}}+o(t).$

Now we give an upper bound of

$|\mu(z)|=\left|\frac{\frac{\alpha}{a}-\frac{\beta}{b}}{1-\frac{\bar{b}}{b}}\right|.$

Assume that $\theta=\arg b$. Then

$|1-\frac{\bar{b}}{b}|=2|\sin\theta|.$

To give an upper bound of the quasiconformal dilatation, we discuss
$\sin\theta$ in two cases.

Let $s_{0}=\sqrt{\frac{4}{\pi}}$. We remark that the area of $|\omega|$ is
$2$. Note that for any edge of $T$, it either has length $\leq 2s_{0}$ or
crosses a maximal flat cylinder $C$ whose height $h$ is greater than its
circumference $c$.

Assume that all edges of $T$ has length $\leq 2s_{0}$. In this case, the
triangle $T$ is inscribed in a circle of radius $d(x,\Sigma)\leq 2s_{0}.$

Since $\sin\theta=|a-b|/2d(x,\Sigma)$, we have

$|\sin\theta|\geq\frac{r}{d(x,\Sigma)}\geq\frac{\sqrt{\pi}r}{4}.$

Thus we have

$\left|\frac{\frac{\alpha}{a}-\frac{\beta}{b}}{1-\frac{\bar{b}}{b}}\right|\leq\frac{8\max\\{|\frac{\alpha}{a}|,|\frac{\alpha}{a}|\\}}{\sqrt{\pi}r}\leq\frac{8}{\sqrt{\pi}r}\|\eta\|_{\mathrm{AGY}}.$

The remaining case is that some edge of $T$ crosses a maximal flat cylinder
$C$ whose height $h$ is greater than its circumference $c$. In this case, some
other edge of $T$ also crosses $C$. Thus the triangle $T$ looks like an
isosceles triangle with a short base. As a result, we may choose the angle
$\theta$ such that

$\frac{\pi}{4}\leq\theta\leq\frac{\pi}{2}.$

Then we have $\sin\theta\geq\frac{\sqrt{2}}{2}$. It follows that

$\left|\frac{\frac{\alpha}{a}-\frac{\beta}{b}}{1-\frac{\bar{b}}{b}}\right|\leq\frac{2\max\\{|\frac{\alpha}{a}|,|\frac{\alpha}{a}|\\}}{\sqrt{2}}\leq\sqrt{2}\|\eta\|_{\mathrm{AGY}}.$

Note that $\pi r^{2}\leq 1$.

∎

######  Remark 3.3.

It is known that for any quadratic differential $q$, in the the direction of
Teichmüller flow, the AGY norm is less than the Teichmüller norm (see [2, Page
152]).

As we have shown in the proof of Proposition 3.2, the order $\frac{1}{r}$
appears when the triangle is almost flat. If there is some angle of the
triangle which is neither close to $0$ or $\pi$, then the Beltrami coefficient
should be bounded above by $\|\eta\|_{\mathrm{AGY}}$ up to a multiplicative
constant.

###  3.3. The order $\frac{1}{r}$ in Proposition 3.2 is sharp.

We recall the following construction of Kahn-Wright [6, §3.3].

Let $\epsilon>0$ be a small constant. We take a square torus of length $1$,
make a length $\epsilon$ horizontal slit, and glue in a small cylinder with
circumference $1$ and height $\epsilon$. The construction defines an Abelian
differential $(X_{\epsilon},\omega_{\epsilon})$ with $1$ double zero.

Figure 1. The examples of Kahn-Wright [6, §3.3].

Let ${\gamma}_{\epsilon}$ be the core curve of the small cylinder on
$(X_{\epsilon},{\omega}_{\epsilon})$. Denote the harmonic differential dual to
$\gamma$ by ${\gamma}_{\epsilon}^{*}$.

######  Remark 3.4.

We can write $\gamma_{\epsilon}^{*}=\beta_{\epsilon}+\bar{\beta}_{\epsilon}$,
where $\beta_{\epsilon}$ is an Abelian differential. It is known that the
Hodge norm of $\beta_{\epsilon}$ is bounded above and below independently of
$\epsilon$.

As shown in Kahn-Wright [6, §3.3],

$\left\|D\pi({\gamma}_{\epsilon}^{*})\right\|_{\mathrm{Teich}}\geq\frac{C}{\epsilon}$

for some constant $C$.

The path $\omega_{\epsilon}+t\epsilon\gamma_{\epsilon}^{*}$ is corresponding
to a family of translations surfaces, obtained by twisting along the core
curve of the small cylinder. When $t=1$,
$\omega_{\epsilon}+\epsilon\gamma_{\epsilon}^{*}$ is a Dehn twist of
$\omega_{\epsilon}$.

The length of shortest saddle connections of $\omega_{\epsilon}$ is equal to
$\epsilon$. If $\alpha_{0}$ is the saddle connection contained in the small
cylinder and crossing $\gamma_{\epsilon}$, then

$\frac{\left|\int_{\alpha_{0}}\gamma_{\epsilon}^{*}\right|}{\left|\int_{\alpha_{0}}\omega_{\epsilon}\right|}=\frac{\epsilon}{\epsilon}=1.$

For any other saddle connections $\alpha$, which crosses the small cylinder
$n$ times, we have

$\frac{\left|\int_{\alpha}\gamma_{\epsilon}^{*}\right|}{\left|\int_{\alpha}\omega_{\epsilon}\right|}\leq\frac{n\epsilon}{n\epsilon}=1.$

As a result, $\|{\gamma}_{\epsilon}^{*}\|_{\mathrm{AGY}}=1$. In conclusion, we
have

$\left\|D\pi({\gamma}_{\epsilon}^{*})\right\|_{\mathrm{Teich}}\geq
C\frac{\|{\gamma}_{\epsilon}^{*}\|_{\mathrm{AGY}}}{\epsilon},$

where $\epsilon$ is the length of shortest saddle connections of
$\omega_{\epsilon}$.

##  4\. The lower bound

In this section, we consider tangent vectors to $\mathcal{P}_{g}$ of the form
$\eta=\bar{\beta}$, where $\beta\in H_{-1}^{1,0}(\hat{X})$. By Theorem 2.1,
the Beltrami differential $\mu=\bar{\beta}/\omega$ can be considered as the
tangent vector $D\pi(\eta)$ via pairing with holomorphic quadratic
differentials:

$\int_{\hat{X}}\rho^{*}(\phi)\frac{\bar{\beta}}{\omega}.$

The Hodge norm of $\beta\in H_{-1}^{1,0}(\hat{X})$ is defined by

$\|\beta\|_{\mathrm{Hodge}}=\sqrt{\int_{\hat{X}}|\beta|^{2}}.$

We have (see [6, Theorem 3.1]):

######  Theorem 4.1.

For any $\eta=\bar{\beta}\in H_{-1}^{0,1}(\hat{X})$, we have

$\|D\pi(\eta)\|_{\mathrm{Teich}}\geq\frac{\|\beta\|_{\mathrm{Hodge}}}{\|\omega\|_{\mathrm{Hodge}}}\
.$

###### The lower bound in Theorem 1.1.

Note that $\|q\|=1$ implies $\|\omega\|_{\mathrm{Hodge}}=\sqrt{2}.$ Applying
Theorem 4.1 and the next proposition , we have

(2) $\displaystyle\|D\pi(\eta)\|_{\mathrm{Teich}}$ $\displaystyle\geq$
$\displaystyle\frac{\|[\eta]\|_{\mathrm{Hodge}}}{\sqrt{2}}\geq\frac{r}{\sqrt{2}}\|\eta\|_{\mathrm{AGY}}.$

∎

######  Proposition 4.2.

Let $2r$ be the shortest length of saddle connections. For any saddle
connection $\gamma$ of $\omega$ and any $\beta\in H_{-1}^{1,0}(\hat{X})$, we
have

$\frac{\left|\int_{\gamma}\beta\right|}{\left|\int_{\gamma}\omega\right|}\leq\frac{\|\beta\|_{\mathrm{Hodge}}}{r}.$

###### Proof.

Endow the surface with the metric $|\omega|$. Let $a,b$ be the zeros of
$\omega$ which are connected by $\gamma$. Let $D_{a}$ and $D_{b}$ be the disks
of radius $r$ around $a$ and $b$. We separate $\gamma$ into two parts by
setting $\gamma^{\prime}=\gamma\cap\left(D_{a}\cup D_{b}\right)$ and
$\gamma^{\prime\prime}=\gamma\setminus\gamma^{\prime}$.

The next inequality is a corollary of [6, Lemma 3.2]:

(3) $\left|\int_{\gamma^{\prime}}\beta\right|\leq
2\|\beta\|_{\mathrm{Hodge}}.$

On the other hand, we have

$\displaystyle\left|\int_{\gamma^{\prime\prime}}\beta\right|\leq\int_{\gamma^{\prime\prime}}\left|\frac{\beta}{\omega}\right|\left|\omega\right|.$

Since $\gamma^{\prime\prime}$ is contained in a saddle connection, we have
$\left|\int_{\gamma^{\prime\prime}}\omega\right|=\int_{\gamma^{\prime\prime}}|\omega|$.
Thus

$\displaystyle\frac{\left|\int_{\gamma^{\prime\prime}}\beta\right|}{\left|\int_{\gamma^{\prime\prime}}\omega\right|}$
$\displaystyle\leq$
$\displaystyle\max_{\gamma^{\prime\prime}}\left|\frac{\beta}{\omega}\right|$
$\displaystyle\leq$
$\displaystyle\frac{\|\beta\|_{\mathrm{Hodge}}}{\sqrt{\pi}r},$

where the last inequality was proved by Kahn-Wright in §3.2 of [6]. Combining
the above inequality with (3), we have

$\displaystyle\frac{\left|\int_{\gamma}\beta\right|}{\left|\int_{\gamma}\omega\right|}$
$\displaystyle\leq$
$\displaystyle\frac{\left|\int_{\gamma^{\prime}}\beta\right|+\left|\int_{\gamma^{\prime\prime}}\beta\right|}{\left|\int_{\gamma^{\prime}}\omega\right|+\left|\int_{\gamma^{\prime\prime}}\omega\right|}$
$\displaystyle\leq$
$\displaystyle\max\left\\{\frac{\left|\int_{\gamma^{\prime}}\beta\right|}{\left|\int_{\gamma^{\prime}}\omega\right|},\frac{\left|\int_{\gamma^{\prime\prime}}\beta\right|}{\left|\int_{\gamma^{\prime\prime}}\omega\right|}\right\\}$
$\displaystyle\leq$
$\displaystyle\max\left\\{\frac{\|\beta\|_{\mathrm{Hodge}}}{r},\frac{\|\beta\|_{\mathrm{Hodge}}}{\sqrt{\pi}r}\right\\}$
$\displaystyle=$ $\displaystyle\frac{\|\beta\|_{\mathrm{Hodge}}}{r}.$

∎

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# Electronic states in a quantum Beltrami surface

J. Furtado111E-mail<EMAIL_ADDRESS>Centro de Ciências e Tecnologias,
Universidade Federal do Cariri, 63048-080, Juazeiro do Norte, Brazil

(August 30, 2024)

###### Abstract

In this paper, we investigate the influence of the geometry in the electronic
states of a quantum Beltrami surface. We have considered an electron governed
by the spinless stationary Schrödinger equation constrained to move on the
Beltrami surface due to a confining potential from which the Da Costa
potential emerges. We investigate the role played by the geometry and orbital
angular momentum on the electronic states of the system.

##  I Introduction

Quantum mechanics at curved surfaces brings about fascinating phenomena.
Quantum particles exhibit wave-like properties, such as diffraction and
interference, even on curved surfaces. When dealing with curved surfaces, such
as the surface of a torus GomesSilva:2020fxo , catenoid euclides , among
others, the principles of quantum mechanics take on a fascinating twist. The
curvature of the surface introduces a geometric factor that significantly
influences the behavior of quantum particles. These phenomena give rise to
intriguing effects, such as the formation of quantized energy levels and the
emergence of geometric phases costa ; costa1 . Additionally, the curvature of
the surface can induce curvature-induced forces on the particles, influencing
their trajectories and dynamics costa ; costa1 . Exploring the interplay
between quantum mechanics and curved surfaces unveils a rich landscape of
phenomena, providing insights into fundamental aspects of both quantum theory
and geometry.

Important two-dimensional nanostructures in low energy physics, such as
graphene katsnelson ; geim ; castro  and phosphorene phosphorene , have
attracted attention due to their unusual properties. The elec tronic
properties of such two-dimensional systems are highly dependent on the
geometry CostaFilho:2020sbw ; Aguiar:2020dgi , so that they can be used as
analog models for high energy physics systems Capozziello:2020ncr ;
Cvetic:2012vg ; Pourhassan:2018wjg ; Acquaviva:2022yiq ; Iorio:2013ifa ;
Iorio:2010pv ; Iorio:2014pwa .

Among the studied surfaces we may highlight the torus. On the toroidal
geometry the curvature effects plays a very significant role
GomesSilva:2020fxo ; encinosa . Carbon nanotori are present in
nanoelectronics, biosensors and quantum computing, Goldsmith ; Goldsmith2 .
Considering an electron with dynamic governed by the Schrödinger equation, the
curvature-induced bound-state eigenvalues and eigenfunctions were calculated
in encinosa  for a particle constrained to move on a toroidal surface. These
same considerations were considered and the action of external fields was
investigated in GomesSilva:2020fxo . Charged spin 1/2 particle, governed by
Pauli equation, moving along a surface of a torus was studied in AGM .
Analytical solutions for the (2 + 1) Dirac equation on the torus surface were
first obtained in Yesiltas:2018zoy  using a supersymmetric approach of quantum
mechanics for two cases, constant and position-dependent Fermi velocity. These
same considerations were taken into account for the investigation of the Dirac
equation on the torus under the action of external fields in Yesiltas:2021crm
. More recently, a possible qubit encoding in the energy levels of a graphene
nanotorus was investigated in Furtado:2022uvk .

The catenoid geometry is drawing a lot of attention in the last years. As a
minimal surface, some two dimensional wormhole geometries are equivalent to
catenoid shaped surfaces dandoloff . In Ref. gonzalez ; pincak  the authors
proposed a bridge connecting a bilayer graphene using a nanotube. In order to
obtain a smooth bridge, Ref.dandoloff ; dandoloff2 , suggested a catenoid
surface aiming to describe the bilayer and the bridge using a single surface.
The catenoid curvature is concentrated around the bridge and vanishes
asymptotically spivak . For non-relativistic electrons, the surface curvature
gives rise to a geometric potential in the Schrödinger equation. The effects
of the external electric and magnetic fields and geometry upon the graphene
catenoid bridge was investigated in Ref.euclides , where it was considered a
single electron whose dynamics is governed by the Schrödinger equation on the
surface. Also, the consideration of a position-dependent mass problem upon the
electron on a catenoid bridge was addressed in Ref. Yesiltas:2021dpm , where
it was proposed an isotropic position-dependent mass as a function of the
Gaussian and mean curvatures. And more recently, the electronic states of the
generalized Ellis-Bronnikov bilayer graphene wormhole were studied
deSouza:2022ioq .

As far as we know, only a few works were carried out regarding the study of
the properties of a quantum Beltrami surface, see f.e. Morresi ; Gallerati .
In Morresi , the authors investigate Event Horizons and Hawking radiation
through a Beltrami shaped graphene membrane. In Gallerati  the author study
the electronic properties of Beltrami shaped graphene-like material where
analytical solutions were obtained for charge carriers described by the
massless Dirac equation. The Beltrami surface is a special type of surface
that possesses constant Gaussian curvature and exhibits interesting geometric
properties. Hence, in this paper, we investigate the influence of the geometry
in the electronic states of a quantum Beltrami surface. We have considered an
electron governed by the spinless stationary Schrödinger equation constrained
to move on the surface of a pseudosphere due to a confining potential from
which the Da Costa potential emerges. We investigate the role played by the
geometry and orbital angular momentum on the electronic states of the system.

This paper is organized as follows: In section II we present the dynamics of
an electron governed by the Schrödinger equation constrained to move on a
Beltrami surface. In section III we calculate the bound states for the
electron on the Beltrami surface considering both the geometric and
centrifugal effects. In section IV we draw our conclusions.

##  II Electron on a Beltrami surface

(a)

(b)

Figure 1: Beltrami surface and coordinate system.

In this section, we introduce the geometry and the dynamics of the electron on
the Beltrami surface considering the effects of the curvature and geometry of
the system. We will be considering an electron described by a quadratic
dispersion relation, so that we employee the Schrödinger equation in curved
surfaces in order to properly describe the system. As shown in fig.1, the
Beltrami surface is realized as the revolution surface generated by the
rotation of the tractrix curve about its asymptote. Our coordinate system is
depicted in fig.1.

After squeezing the electron wave-function on the surface, the spinless
stationary Schrödinger equation has the form

$-\frac{\hbar^{2}}{2m^{*}}\nabla^{2}\Psi+V_{dc}\Psi=E\Psi,$ (1)

where
$\nabla^{2}\Psi=\frac{1}{\sqrt{g}}\partial_{a}(\sqrt{g}g^{ab}\partial_{b}\Psi)$
is the Laplacian operator on the surface, $g^{ab}$ is the induced metric of
the surface, $m^{*}$ is the effective mass of the electron and
$V_{dC}=-\frac{\hbar^{2}}{2m^{*}}(H^{2}-K)$ is a potential induced by the
surface curvature, known as the da Costa potential costa . The geometric
potential depends both on the the mean curvature $H$ and the Gaussian
curvature $K$ costa .

We adopt the Beltrami coordinates as

$\displaystyle\vec{r}(u,\phi)$ $\displaystyle=$ $\displaystyle
R\operatorname{sech}\left(\frac{u}{R}\right)\cos(\phi)\hat{i}+R\operatorname{sech}\left(\frac{u}{R}\right)\sin(\phi)\hat{j}+$
(2) $\displaystyle+\left(u-R\tanh\left(\frac{u}{R}\right)\right)\hat{k},$

where $R$ is the radius of the pseudosphere in $u=0$. The coordinates
$u\in(-\infty,\infty)$ and $\phi\in[0,2\pi)$ cover the whole pseudosphere. It
is important to highlight that for a real two-dimensional material, some
cutoffs in the $u$-coordinate must be imposed in order to properly guarantee
the validity of the continuum limit employed here. For a graphene-like
material, for example, the bonds in the graphene lattice are around $1,43\AA$
graphene-bond , so that the radius $r$ for a given value of $u$ must be
$r>>1,43\AA$ in order to ensure the continuum limit approach.

Figure 2: Da Costa potential for the pseudosphere.

In this coordinate system, the interval reads

$\displaystyle
ds^{2}=\tanh^{2}\left(\frac{u}{R}\right)du^{2}+R^{2}\operatorname{sech}^{2}\left(\frac{u}{R}\right)d\phi^{2},$
(3)

and then, the diagonal components of the induced metric tensor on the Beltrami
surface are

$\displaystyle g_{uu}$ $\displaystyle=$
$\displaystyle\tanh^{2}\left(\frac{u}{R}\right),$ (4) $\displaystyle
g_{\phi\phi}$ $\displaystyle=$ $\displaystyle
R^{2}\operatorname{sech}^{2}\left(\frac{u}{R}\right).$ (5)

The nonvanishing components of the Christoffel symbols are straightforwardly
calculated and written as

$\displaystyle\Gamma^{u}_{uu}$ $\displaystyle=$
$\displaystyle\frac{2\operatorname{csch}\left(\frac{2u}{R}\right)}{R},$ (6)
$\displaystyle\Gamma^{u}_{\phi\phi}$ $\displaystyle=$ $\displaystyle
2R\operatorname{csch}\left(\frac{2u}{R}\right),$ (7)
$\displaystyle\Gamma^{\phi}_{u\phi}$ $\displaystyle=$
$\displaystyle\Gamma^{\phi}_{\phi
u}=-\frac{\tanh\left(\frac{u}{R}\right)}{R}.$ (8)

In order to properly take into account the effects of the geometry in the
dynamics of a non-relativistic electron constrained to move on the Beltrami
surface, we must compute the Da Costa potential, which is given by

$\displaystyle
V_{dC}(u)=-\frac{\hbar^{2}}{8m^{*}R^{2}}\cosh^{2}\left(\frac{u}{R}\right)\coth^{2}\left(\frac{u}{R}\right).$
(9)

It is worthwhile to mention that the da Costa potential $V_{dc}$ exhibits a
parity-symmetrical potential well with respect to $u=0$. Such a well tends to
trap the electron in a ring around $u=0$, as we can see in Fig.2. In the limit
when $u\rightarrow 0$ the da Costa potential $V_{dC}\rightarrow-\infty$, such
behaviour occurs due to the non differentiability of the surface in $u=0$.
Hence, the da Costa potential tends to confine the particle around $u=0$ due
to the edge in the surface. This behaviour is depicted in fig. (2) for three
values of $R$, namely, $R=1$, $R=10$ and $R=100$. An important point that must
be highlighted here is that the inclusion of the da Costa’s potential is not
in disagreement with the results found in Liu . In Liu  the authors find that
there is no geometric potential for a Dirac fermion on a two-dimensional
curved surface of revolution. However, we are considering here a quadratic
dispersion relation for the charge carriers, so that they are described by the
Schrödinger equation instead of Dirac equation. Hence, we must consider the da
Costa’s potential.

The axial symmetry leads to the periodic behavior of the wave function in the
form

$\Psi(u,\phi)=\psi(u)e^{i\ell\phi},$ (10)

where $\ell$ is the orbital quantum number. Substituting the Eq.(10) into
Eq.(1), the stationary Schrödinger equation becomes

$\displaystyle-\frac{\hbar^{2}}{2m}\Lambda_{1}(u)\frac{d^{2}\phi(u)}{du^{2}}+\Lambda_{2}(u)\frac{d\phi(u)}{du}+\Lambda_{3}(u)\phi(u)=E\phi(u),$
(11)

with $\Lambda_{1}(u)$, $\Lambda_{2}(u)$ and $\Lambda_{3}(u)$ given by:

$\displaystyle\Lambda_{1}(u)$ $\displaystyle=$
$\displaystyle\coth^{2}\left(\frac{u}{R}\right)$ (12)
$\displaystyle\Lambda_{2}(u)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2mR}\coth^{3}\left(\frac{u}{R}\right)$ (13)
$\displaystyle\Lambda_{3}(u)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{8mR^{2}}\cosh^{2}\left(\frac{u}{R}\right)\left(4l^{2}-\coth^{2}\left(\frac{u}{R}\right)\right)$
(14)

Note that the Eq.11 exhibits parity and time-reversal invariance, as a result
of the Beltrami surface geometric symmetries. Nonetheless, the first order
derivative terms render the Hamiltonian non-Hermitian with respect to the
momentum $\hat{P}_{u}:=-i\hbar\partial_{u}$. The non-Hermiticity of the free
electron Hamiltonian is not a problem, since the space-time reflection
symmetry is preserved, the spectrum of the eigenvalues of the Hamiltonian is
completely real Bender ; Bender2 . Besides, there is an Hermitian equivalent
Hamiltonian that can be achieved by a simple changing of variables.
Considering the change in the wave function given by

$\phi(u)=\sqrt{\cosh\left(\frac{u}{R}\right)\tanh\left(\frac{u}{R}\right)}y(u)$
(15)

we are lead to the following one dimensional Hermitian Schrödinger-like
equation

$\Lambda_{1}(u)\frac{d^{2}y(u)}{du^{2}}+V_{eff}(u)y(u)=Ey(u),$ (16)

where

$\displaystyle
V_{eff}=\frac{\hbar^{2}}{16mR^{2}}\left[\cosh\left(\frac{2u}{R}\right)+1\right]\left[4l^{2}+3\operatorname{csch}^{4}\left(\frac{u}{R}\right)-1\right].$
(17)

At this point we must highlight an important feature that becomes evident from
the effective potential presented above, which is the fact that there is no
break of chirality since the dependence on the orbital angular momentum is
squared.

In order to write our dynamic equation in a more convenient way, let us
perform the following transformation in the wave function, i.e.,
$y(u)=s(u)X(u)$, where $s(u)$ is given by
$s(u)=e^{\frac{1}{2}\coth^{2}\left(\frac{u}{R}\right)}$. By considering such
transformation we are able to write the following equation

$-\frac{\hbar^{2}}{2}\frac{d}{du}\left[\frac{\Lambda_{1}(u)}{m^{*}}\frac{dX(u)}{du}\right]+\bar{V}_{eff}(u)X(u)=EX(u),$
(18)

where $\bar{V}_{eff}(u)$ is written as

$\bar{V}_{eff}(u)=V_{eff}(u)-\frac{\hbar^{2}}{2}\frac{\Lambda_{1}(u)}{m^{*}}\left[\frac{s^{\prime\prime}(u)}{s(u)}\right].$
(19)

Notice that the Schrödinger equation as written in eq.(18), can be identified
as describing a system whose mass is position-dependent pnbilayer ; sinner ;
pdm1 ; pdm2 ; pdm3 ; pdm4 ; moraes . In the present case, we can identify the
position-dependent mass as being

$M(u)=\frac{m^{*}}{\Lambda_{1}(u)}.$ (20)

Such position-dependent mass exhibit an asymptotic behaviour so that when
$u\rightarrow\pm\infty$ we have $M(u)\rightarrow m^{*}$. Also, when
$u\rightarrow 0$, the effective mass reaches its minimum value. Such behaviour
is a consequence of the singularity at $u=0$ and could be an indication that
at $u=0$ the mean distance of the sites in the lattice structure of the
material is greater in comparison to other regions of the surface. A greater
mean distance between the sites of the lattice yields a smaller interaction
with the electrons of the surface, which leads to a smaller effective mass.
This is in agreement with more phenomenological considerations based on the
Heisenberg uncertainty principle and the stretching of the manifold due to
curvature, see f.e. dandoloff2 ; atanasov22 . The behaviour of the position-
dependent mass is depicted in fig.(3) for $m^{*}=1$ and three values of $R$,
namely, $R=1$, $R=5$ and $R=10$.

Figure 3: Behaviour of the position-dependent mass for three values of $R$,
namely, $R=1$, $R=5$, $R=10$ and $m^{*}=1$.

###  II.1 Qualitative analysis

Before obtaining the bound states and their respective spectra, let us discuss
some qualitative features of the effective potential.

In the fig.4, we show the effective potential for three values of $R$, namely,
$R=1$, $R=5$ and $R=10$ and orbital angular momentum $\ell=0$, obtained by the
eq.19. We can see that the effective potential exhibit a behaviour
qualitatively similar to the Da Costa potential presented in fig.(2). The same
divergence is present at $u=0$ as an edge-like effect. An important feature
that can be observed in fig.4 is the fact that by increasing the value of $R$
the infinite well centered at $u=0$ becomes wider.

Figure 4: The effective potential for three values of $R$, namely, $R=1$,
$R=5$ and $R=10$ and $\ell=0$, obtained by the eq.19. Figure 5: The effective
potential for $R=1$ and three values of the orbital angular momentum, namely,
$\ell=0$, $\ell=5$ and $\ell=10$, obtained by the eq.19.

The effective potential for $R=1$ and $\ell=0$, $\ell=5$ and $\ell=10$ is
shown in the fig.5. In this figure, in addition to the geometric influence on
the electron, there is the centrifugal potential emergent from the orbital
angular momentum that significantly modifies the effective potential. When
comparing the figures 4 and 5, we observe that the centrifugal effect is
suppressed for regions close to $u=0$. However, when we move away from $u=0$,
the influence of the orbital angular momentum becomes more relevant and the
effective potential starts to exhibit a prominent growth due to the dominance
of the orbital angular momentum over the geometric effects.

##  III Bound states

Let us investigate now the influence of both geometry and orbital angular
momentum on the electronic states on the quantum Beltrami surface. For
pedagogical purposes, let us study first the case with no orbital angular
momentum, i.e., $\ell=0$ for three values of radius $R$, namely, $R=1$, $R=10$
and $R=20$. We have considered for all the plots $\hbar=1$ and $m^{*}=1$.

In fig. (6a) we are considering $R=1$ and no orbital angular momentum. For
this configuration we can see that the first eigenvalues of the energy are not
bound states, but propagating states. The infinite well centered at $u=0$ for
$R=1$ is not wide enough to allow bound states. In a certain sense we can
interpret this configuration as a reflectionless shaped potential.

The fig. (6b) exhibit the first two states for $R=10$ and $\ell=0$. As we can
see these bound states are practically degenerated and they are weakly
confined at the well at $u=0$. But it is already possible to infer that by
increasing the value of the radius $R$ more confined states become allowed.
Finally, when we consider $R=20$ and $\ell=0$, as depicted in (6c), we can see
a more confined bound state, confirming that as we increase the value of $R$
we allow the existence of more confined bound states.

(a)

(b)

(c)

Figure 6: Eigenenergies for the position-dependent Schrödinger equation for
three values of $R$, namely, $R=1$, $R=10$ and $R=20$. For this plot we have
disregarded the orbital angular momentum. We have considered $\hbar=1$ and
$m^{*}=1$.

In the figure (7) we have considered the effect of orbital angular momentum in
the electronic states of the Beltrami surface. In order to do so, we have
considered $R=1$ and three values of the orbital angular momentum, namely,
$\ell=0$, $\ell=5$ and $\ell=10$. Also, for the plots we have considered
$\hbar=1$ and $m^{*}=1$. In this plot we have added the case where $R=1$ and
$\ell=0$, already discussed previously for the sake of comparison.

In fig (7b) we have considered $R=1$ and orbital angular momentum $\ell=5$.
Different from the case when $\ell=0$, as already discussed, the effect of the
orbital angular momentum gives rise to a centrifugal potential that becomes
dominant as we move away from $u=0$. The well formed due to the orbital
angular momentum allows the existence of bound states. The first five bound
states are depicted in (7b). It is important to highlight here that the first
two bound states are degenerated (red line), as well as the third and fourth
bound states.

Finally, for $R=1$ and $\ell=10$ we have a similar behaviour in comparison to
the case with $\ell=5$. The main difference between the two cases is the
energy scale. For this case also, the well formed by the orbital angular
momentum provides the possibility of bound states. The first two bound states
(red line) are degenerated as well as the third and fourth bound state.

It is worthwhile to mention that the gaps in the eigenenergies $\Delta_{1}$
and $\Delta_{2}$, presented in (7b) are different, i.e.,
$\Delta_{1}\neq\Delta_{2}$. Such feature makes our system entirely different
from a quantum harmonic oscillator, in which all the gaps between the
eigenenergies are equal to $\hbar\omega$. This annharmonicity is an indication
that the present system can fulfill the Di Vicenzo’s criteria, which means
that our system could be thought as a qubit suitable for encoding quantum
information vicenzo . Such a remarkable feature will be discussed in more
details in a future paper.

(a)

(b)

(c)

Figure 7: Eigenenergies for the position-dependent Schrödinger equation for
$R=1$ and three values of orbital angular momentum $\ell$, namely, $\ell=0$,
$\ell=5$ and $\ell=10$. For this plot we have considered $\hbar=1$ and
$m^{*}=1$.

##  IV Final Remarks

In this paper, we investigate the influence of the geometry in the electronic
states of a quantum Beltrami surface. We have considered an electron governed
by the spinless stationary Schrödinger equation constrained to move on the
surface of a pseudosphere due to a confining potential from which the Da Costa
potential emerges. We investigate the role played by the geometry and orbital
angular momentum on the electronic states of the system.

Initially we describe the geometry of the Beltrami surface and we show how the
Da Costa potential, which is constructed by the mean and Gaussian curvatures,
exhibit an infinite well at $u=0$ as some kind of edge effect. The Hamiltonian
governing the electron dynamics was obtained directly from the geometry and it
is non-Hermitian but $\mathcal{PT}$ symmetric. The invariance under parity
reversion and time inversion allowed us to obtain an Hermitian equivalent
Hamiltonian.

The electron on the Beltrami surface behaves as if its mass depends on the
position. The mass of the electron is smaller at $u=0$, also a consequence of
the edge at $u=0$. The position dependency of the mass lead us to speculate
that the effects of the lattice strain, which emerges as a consequence of the
curvature, increase the mean distance of the sites of the lattice at $u=0$,
which in its turn yield to a smaller mass at $u=0$.

We have shown that the effective potential in the absence of orbital angular
momentum exhibit a behaviour qualitatively similar to the Da Costa potential.
The same divergence is present at $u=0$ as an edge-like effect. An important
feature that could be observed is the fact that by increasing the value of $R$
the infinite well centered at $u=0$ becomes wider.

By considering the effects of orbital angular momentum we observe that the
centrifugal effect is suppressed for regions close to u = 0. However, when we
move away from $u=0$, the influence of the orbital angular momentum becomes
more relevant and the effective potential starts to exhibit a prominent growth
due to the dominance of the orbital angular momentum over the geometric
effects.

The bound state were numerically computed for the cases with and without
orbital angular momentum. For the cases with no contribution of the orbital
angular momentum, we could see that as we increase the value of the Beltrami
radius $R$ we obtain more confined bound states. The consideration of a non-
vanishing orbital angular momentum allows the existence of more confined bound
states whose energy depends on the ratio between $R$ and $\ell$. An important
feature that must be highlighted is the fact that there is an annharmonicity
in the Hamiltonian of the system for a non-vanishing $\ell$. Such
annharmonicity may lead us to the possibility of enconding a qubit in the
energy states of the system. Such an important feature will be addressed in a
future work.

##  V Acknowledgments*

The author would like to thank J.E.G. Silva for valuable discussions,
Alexandra Elbakyan and Sci-Hub, for removing all barriers in the way of
science and the Fundação Cearense de Apoio ao Desenvolvimento Científico e
Tecnológico (FUNCAP) under the grant PRONEM PNE0112- 00085.01.00/16.

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