# On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

Authors: Yan Wu; Yi Qi; Zunwei Fu
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

LetA
Z
(
R
) be the infinitesimal asymptotic Teichmüller space of a Riemann surface R of infinite type. It is known that A
Z
(
R
) is the quotient Banach space of the infinitesimal Teichmüller space Z
(
R
), where Z
(
R
) is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points in A
Z
(
R
). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller space A
Z
(
D
) by constructing a special degenerating sequence.


---

## Body

## 1. Introduction

LetR be a hyperbolic Riemann surface, that is, a Riemann surface with universal covering surface which is conformally equivalent to the open unit disk D. Denote by L
∞
(
R
) the Banach space of Beltrami differentials μ
=
μ
(
z
)
d
z
¯
/
d
z on X with finite L
∞-norms.LetQ
(
R
) be the space of integrable holomorphic quadratic differentials ϕ
=
ϕ
(
z
)
d
z
2 on R with L
1-norms: (1)
ϕ
=
∫
R
ϕ
z
d
x
d
y
<
+
∞
.Denote by Q
1
(
R
) the unit sphere of Q
(
R
) and by Q
d
(
R
) the set of all degenerating sequences of Q
(
R
). A sequence of quadratic differentials {
ϕ
n
} in Q
(
R
) is said to be a degenerating sequence if ϕ
n
=
1 and ϕ
n
→
0 locally uniformly on R as n
→
∞.Two elementsμ
,
ν
∈
L
∞
(
R
) are called infinitesimally Teichmüller equivalent, if (2)
∫
R
μ
ϕ
=
∫
R
ν
ϕ
∀
ϕ
∈
Q
R
.We denote by [
μ
]
z the infinitesimal Teichmüller equivalence class of μ. The infinitesimal Teichmüller space (3)
Z
R
≔
μ
z
:
μ
∈
L
∞
Ris the set of all infinitesimal Teichmüller equivalence classes of μ’s in L
∞
(
R
). The point [
0
]
z is called the basepoint of Z
(
R
).A Beltrami differentialμ
∈
L
∞
(
R
) is said to be vanishing at infinity, if, for each ϵ
>
0, there is a compact set E in R such that μ
R
-
E
∞
<
ϵ. Denote by L
0
∞
(
R
) the set of all such vanishing μ’s and by (4)
Z
0
R
≔
μ
z
∈
Z
R
:
μ
∈
L
0
∞the set of all infinitesimal Teichmüller equivalence classes of μ’s in L
0
∞
(
R
).Two elementsμ
,
ν
∈
L
∞
(
R
) are called infinitesimally asymptotically equivalent, if there exists σ
∈
L
0
∞
(
R
) such that (5)
∫
R
μ
ϕ
=
∫
R
ν
ϕ
+
∫
R
σ
ϕ
∀
ϕ
∈
Q
R
.We denote by [
[
μ
]
]
z the infinitesimal asymptotic equivalence class of μ. The infinitesimal asymptotic Teichmüller space (6)
A
Z
R
≔
μ
z
:
μ
∈
L
∞
Ris the set of all infinitesimal asymptotic equivalence classes of μ’s in L
∞
(
R
). The point [
[
0
]
]
z is called the basepoint of A
Z
(
R
).It is known thatZ
(
R
) and A
Z
(
R
) are the tangent spaces to the classical Teichmüller space and asymptotic Teichmüller space at their basepoint, respectively (see [1]). For further results and properties about Teichmüller theory, we refer to the papers [1–5] and the books [6–8].The notion of geodesics plays an important role in the study of the geometry of Teichmüller theory. A geodesic in a metric space is a continuous curve such that for any subarc its length is equal to the distance between its two endpoints. The existence and uniqueness of geodesic between two points in various spaces have been discussed for a long time (see [6, 9–15]). Given two points in infinitesimal Teichmüller space Z
(
R
), it is shown in [6] that there exists at least one geodesic joining them. Furthermore, it is proved in [14] that there exists precisely one geodesic segment joining the basepoint [
0
]
z and [
μ
]
z in Z
(
R
) if and only if [
μ
]
z contains a uniquely extremal Beltrami coefficient with constant modulus.The existence of geodesic joining two given points inA
Z
(
R
) has been proved in [1, 6]; however, the uniqueness of geodesic is unknown. In this paper, we will prove the nonuniqueness of geodesics in the universal infinitesimal asymptotic Teichmüller space A
Z
(
D
) by constructing a special degenerating sequence {
ϕ
n
} in Q
(
D
).The structure of this paper is as follows. Section2 is devoted to setting up the notations and some results we need. In Section 3, a special sequence {
ϕ
n
} degenerating towards a boundary point of the unit circle is constructed. In Section 4, it is proved that there are infinitely many geodesics joining [
[
μ
]
]
z with the basepoint in A
Z
(
D
) when [
[
μ
]
]
z is of a nonsubstantial point using the important lemma (Lemma 2) and the constructed sequence in the previous section.


## 2. Preliminaries

In this section, we recall some notions and basic results from the Teichmüller theory. For more details, we refer to the book [6] and the paper [1].By the Hahn-Banach and Riesz representation theorem, every elementV in the dual space Q
∗
(
R
) of the Banach space Q
(
R
) of all integrable holomorphic quadratic differentials can be represented as (7)
V
ϕ
=
∫
R
μ
ϕ
,
ϕ
∈
Q
R
,where μ is a Beltrami differential in L
∞
(
R
). So there is natural one-to-one correspondence between the infinitesimal Teichmüller space Z
(
R
) and the dual space Q
∗
(
R
). Thus, in what follows, the infinitesimal Teichmüller equivalence classes of Beltrami differentials on R and complex linear functionals on Q
(
R
) are used for points of Z
(
R
) alternately.For everyV
∈
Z
(
R
), the infinitesimal extremal dilatation and the infinitesimal boundary dilatation of V are defined as (8)
V
=
μ
z
=
inf
⁡
μ
∞
:
μ
r
e
p
r
e
s
e
n
t
s
V
,
(9)
b
V
=
b
μ
z
=
inf
⁡
b
∗
μ
:
μ
r
e
p
r
e
s
e
n
t
s
V
,respectively, where (10)
b
∗
μ
=
inf
⁡
μ
R
-
E
∞
:
E
i
s
a
c
o
m
p
a
c
t
s
u
b
s
e
t
o
f
R
.
μ is called infinitesimal extremal if μ
∞
=
V. It is shown in [6] that Z
(
R
) is a Banach space with the infinitesimal Teichmüller norm (11)
V
=
s
u
p
ϕ
∈
Q
1
R
∫
R
μ
ϕ
=
inf
⁡
b
∗
μ
:
μ
r
e
p
r
e
s
e
n
t
s
V
.The infinitesimal Teichmüller distance between two pointsV
1 and V
2 in Z
(
R
) is defined in [14] (12)
d
V
1
,
V
2
=
V
1
-
V
2
=
s
u
p
ϕ
∈
Q
1
R
∫
R
μ
1
-
μ
2
ϕ
,where μ
1 and μ
2 represent V
1 and V
2, respectively.LetP
z
:
Z
(
R
)
→
A
Z
(
R
); V
↦
V
^ be the quotient mapping from the tangent space Z
(
R
) to the tangent space A
Z
(
R
). V
1 and V
2 in Z
(
R
) represent the same point V
^ in A
Z
(
R
) if V
1
-
V
2
∈
Z
0
(
R
). Thus, the infinitesimally asymptotic equivalence classes [
[
μ
]
]
z of Beltrami differentials are in one-to-one correspondence with the elements V
^ of A
Z
(
R
) and μ represents V
^.For anyV
^
∈
A
Z
(
R
), we define the quotient norm on the quotient space A
Z
(
R
) as (13)
b
V
^
=
b
μ
z
=
inf
⁡
b
∗
μ
:
μ
r
e
p
r
e
s
e
n
t
s
V
^
.It is known in [6] that b
(
V
^
)
=
b
(
V
). μ is called infinitesimally asymptotically extremal if b
∗
(
μ
)
=
b
(
V
^
). Furthermore, A
Z
(
R
) is a Banach space with the standard seminorm (see [6]) (14)
b
V
^
=
s
u
p
ϕ
n
∈
Q
d
R
l
i
m
s
u
p
n
→
∞
V
ϕ
n
=
s
u
p
ϕ
n
∈
Q
d
R
l
i
m
s
u
p
n
→
∞
∫
R
μ
ϕ
n
,where μ represents V
^. In particular, if V
^
∈
Z
0
(
R
), it holds (15)
b
V
^
=
s
u
p
ϕ
n
∈
Q
d
R
l
i
m
s
u
p
n
→
∞
∫
R
μ
ϕ
n
=
0
.The infinitesimal asymptotic Teichmüller distance between two pointsV
^
1 and V
^
2 in A
Z
(
R
) is defined as (16)
d
¯
V
^
1
,
V
^
2
=
b
V
^
1
-
V
^
2
=
s
u
p
ϕ
n
∈
Q
d
R
l
i
m
s
u
p
n
→
∞
∫
R
μ
-
ν
ϕ
n
,where μ and ν represent V
1 and V
2, respectively.For anyV
∈
Z
(
R
), μ
∈
L
∞
(
R
) is an infinitesimal extremal representative of V if and only if it has a so-called Hamilton sequence, namely, a sequence {
ϕ
n
}
∈
Q
1
(
R
), such that (17)
l
i
m
n
→
∞
∫
R
μ
ϕ
n
=
μ
∞
.Similarly, for any V
^
∈
A
Z
(
R
), μ
∈
L
∞
(
R
) is an infinitesimal asymptotically extremal representative of V
^ [1, 6] if and only if there exists an asymptotic Hamilton sequence of μ, namely, a degenerating sequence {
ϕ
n
}
⊂
Q
d
(
R
), such that (18)
l
i
m
n
→
∞
∫
R
μ
ϕ
n
=
b
∗
μ
.LetD
=
{
z
:
|
z
|
<
1
} be the unit disk in the extended complex plane C
^ and let ∂
D be the unit circle. In the following part, we consider some results about the infinitesimal local boundary dilatation of V in the tangent space Z
(
D
) to the universal Teichmüller space.Setp
∈
∂
D and U
r
(
p
)
=
D
∩
z
:
z
-
p
<
r. For any V
∈
Z
(
D
), the infinitesimal local boundary dilatation of V at p is defined as (19)
b
p
V
=
b
p
μ
z
=
inf
⁡
b
p
∗
μ
:
μ
r
e
p
r
e
s
e
n
t
s
V
,where (20)
b
p
∗
μ
=
i
n
f
r
μ
z
U
r
p
∞
:
z
∈
U
r
p
.For anyV
^
∈
A
Z
(
D
), we define (21)
b
p
V
^
=
b
p
μ
z
=
inf
⁡
b
p
∗
μ
:
μ
r
e
p
r
e
s
e
n
t
s
V
^
,and then b
p
(
V
)
=
b
p
(
V
^
). It is proved by Lakic [16] that (22)
b
V
^
=
m
a
x
p
∈
∂
D
b
p
V
^
.A point p
∈
∂
D with b
p
(
V
^
)
=
b
(
V
^
) is said to be an infinitesimal substantial boundary point for V
^. V
^ (or [
[
μ
]
]
z) is called an infinitesimal substantial point in A
Z
(
D
), if every p
∈
∂
D is an infinitesimal substantial boundary point for V
^ (or [
[
μ
]
]
z); otherwise, V
^ (or [
[
μ
]
]
z) is called an infinitesimal nonsubstantial point.The following lemma can be obtained in [17] by Fehlmann and Sakan.Lemma 1.
For anyV
∈
Z
(
D
), let μ
∈
L
∞
(
D
) be an infinitesimal extremal representative of V. Suppose there is a point p
∈
∂
D which is not a substantial boundary point of V. Then, there is an open interval I
⊂
∂
D, p
∈
I, and a domain (23)
S
I
=
z
∈
D
∣
z
≠
0
,
z
z
∈
I
,such that, for any degenerating Hamilton sequence {
ψ
n
} of μ, one has (24)
l
i
m
n
→
∞
∫
S
I
ψ
n
d
x
d
y
=
0
.


## 3. Constructing a Special Sequence Degenerating towards a Boundary Point

In this section, we will construct a special sequence degenerating towards a boundary pointp
∈
∂
D. The method used here is similar to that in [18] while the sequence degenerates towards the whole boundary in this paper.Letp
∈
∂
D and U
r
=
{
z
∈
D
:
|
z
-
p
|
<
r
}. A degenerating sequence {
ϕ
n
}
⊂
Q
d
(
D
) is said to degenerate towards p if, for any neighbourhood U
r of p with r
>
0, (25)
l
i
m
n
→
∞
∫
U
r
p
ϕ
n
z
d
x
d
y
=
1
.Then, for any ϵ
>
0 and r
>
0, there exists a positive integer N, such that (26)
1
-
ϵ
<
∫
U
r
ϕ
n
z
d
x
d
y
<
1holds for every n
>
N. Since ∥
ϕ
n
∥
=
1, ∀
n
∈
N, there exists a positive number r
1
<
1 satisfying (27)
∫
D
∖
D
r
1
ϕ
1
>
1
-
1
2
.By the definition of degenerating sequence and (26), there exists ϕ
k
2 such that (28)
ϕ
k
2
<
1
2
2
,
for
z
≤
1
-
r
1
,
(29)
∫
D
∖
U
r
1
ϕ
k
2
<
1
4
2
.Choose a positive numberr
2
<
r
1 such that (30)
∫
D
∖
U
r
2
ϕ
k
2
>
1
-
1
4
2
,
(31)
∫
U
r
2
ϕ
k
1
<
1
2
2
,where k
1
=
1. It follows from (29) and (30) that (32)
∫
U
r
1
∖
U
r
2
ϕ
k
2
>
1
-
1
2
2
.By induction, we obtain a subsequence{
ϕ
k
n
} of {
ϕ
k
} and a positive number sequence {
r
n
} with r
n
<
r
n
-
1 and l
i
m
n
→
∞
r
n
=
0 such that (33)
ϕ
k
n
<
1
2
n
,
i
n
z
≤
1
-
r
n
,
(34)
∫
U
r
n
-
1
∖
U
r
n
ϕ
k
n
>
1
-
1
2
n
,
(35)
∫
U
r
n
ϕ
k
j
<
1
2
n
,
j
=
1,2
,
…
,
n
-
1
,where n
=
2,3
,
….Without loss of generality, from now on, we writeϕ
n instead of ϕ
k
n for simplicity. Let (36)
ϕ
=
∑
n
=
1
∞
ϕ
n
.From (33), it is easy to see that this series is uniformly convergent in every compact subset of D. So ϕ is a holomorphic quadratic differential on D.Noting thatϕ
n
=
1 and by (34), we get (37)
∫
U
r
n
ϕ
n
≤
1
2
n
,
∫
D
∖
U
r
n
-
1
ϕ
n
≤
1
2
n
;that is, (38)
∫
U
r
n
ϕ
n
=
O
2
-
n
,
n
⟶
∞
,
∫
D
∖
U
r
n
-
1
ϕ
n
=
O
2
-
n
,
n
⟶
∞
.Moreover, by the second formula of (37), we have (39)
∫
D
∖
U
r
n
ϕ
j
≤
1
2
j
,
j
≥
n
+
1
.By simple calculation, it follows from (35), (37), and (39) that (40)
∫
U
r
n
-
1
∖
U
r
n
ϕ
-
ϕ
n
≤
∑
j
=
1
n
-
1
∫
U
r
n
-
1
∖
U
r
n
ϕ
j
+
∑
j
=
n
+
1
∞
∫
U
r
n
-
1
∖
U
r
n
ϕ
j
<
∑
j
=
1
n
-
1
∫
U
r
n
-
1
ϕ
j
+
∑
j
=
n
+
1
∞
∫
D
∖
U
r
n
ϕ
j
<
∑
j
=
1
n
-
1
1
2
n
-
1
+
∑
j
=
n
+
1
∞
1
2
j
<
n
2
n
-
1
.Then, (41)
∫
U
r
n
-
1
∖
U
r
n
ϕ
-
ϕ
n
=
O
n
2
-
n
-
1
,
n
⟶
∞
.Furthermore, from (34) and (41), we have (42)
∫
U
r
n
-
1
∖
U
r
n
ϕ
=
1
+
O
n
2
-
n
-
1
,
n
⟶
∞
.


## 4. Nonuniqueness of Geodesics Joining Every Infinitesimal Nonsubstantial Point with the Basepoint inA
Z
(
D
)

For everyV
^
∈
A
Z
(
R
), it is known in [1] that there exists a representative μ such that μ
∞
=
b
(
V
^
). Then(43)
μ
∞
=
b
V
^
≤
V
≤
μ
∞means μ is an infinitesimal extremal representative of V
^, and (44)
μ
∞
=
b
V
^
≤
b
∗
μ
≤
μ
∞implies μ is an infinitesimal asymptotic extremal representative of V
^. Set b
(
V
^
)
=
k and (45)
γ
t
:
0
,
k
⟶
A
Z
R
;
t
⟼
t
μ
k
z
.It is clear that γ
(
0
)
=
[
[
0
]
]
z and γ
(
k
)
=
[
[
μ
]
]
z; moreover, γ
(
[
0
,
k
]
) is a geodesic segment joining 0 and V
^ in A
Z
(
R
).Letp
∈
∂
D and U
r
(
p
)
=
{
z
∈
D
:
|
z
-
p
|
<
r
}. In this section, we will discuss the nonuniqueness of geodesic segments between 0 and V
^ in A
Z
(
D
) when V
^ is an infinitesimal nonsubstantial point; that is, there exists a point p
∈
∂
D with b
p
(
V
^
)
<
b
(
V
^
).We need the following important lemma.Lemma 2.
LetV
^
∈
A
Z
(
D
) and p
∈
∂
D. Then, for any given ϵ
>
0, there exists an infinitesimal asymptotic extremal representative μ of V
^ such that (46)
b
p
∗
μ
<
b
p
V
^
+
ϵ
.Proof.
Supposep is an infinitesimal substantial boundary point for V
^; that is, b
p
(
V
^
)
=
b
(
V
^
). It is known in [6] that there exists an infinitesimal asymptotic extremal representative μ
∈
L
∞
(
D
) such that b
(
V
^
)
=
b
∗
(
μ
). We conclude that (46) holds since (47)
b
p
∗
μ
≤
b
∗
μ
=
b
V
^
=
b
p
V
^
.
Otherwise, supposeb
p
(
V
^
)
<
b
(
V
^
). For any ϵ
>
0, without loss of generality, we assume that ϵ
<
b
(
V
^
)
-
b
p
(
V
^
). By the definition of the infinitesimal local boundary dilatation, there exists a Beltrami differential μ representing V
^ such that (48)
b
p
∗
μ
<
b
p
V
^
+
ϵ
2
<
b
V
^
.Moreover, by the definition of b
p
∗
(
μ
), there exists r
0
>
0 such that (49)
μ
U
r
0
p
∞
<
b
p
∗
μ
+
ϵ
2
.Let ν be an infinitesimal extremal and asymptotic extremal representative of V
^; that is, ν
∞
=
b
(
V
^
), and let {
ϕ
n
}
⊂
Q
d
(
R
) be an asymptotic Hamilton sequence of ν. By Lemma 1 (r
<
r
0 sufficiently small), we have (50)
l
i
m
n
→
∞
∫
U
r
p
ϕ
n
d
x
d
y
=
0
.So (51)
l
i
m
n
→
∞
∫
U
r
p
ν
ϕ
n
d
x
d
y
≤
ν
∞
l
i
m
n
→
∞
∫
U
r
p
ϕ
n
d
x
d
y
=
0
,which means (52)
l
i
m
n
→
∞
∫
U
r
p
ν
ϕ
n
d
x
d
y
=
0
.
There exists a boundary pointp
′
∈
∂
D outside U
r
(
p
) satisfying (53)
b
p
′
V
^
=
b
V
^
,so we have b
p
′
∗
(
v
)
=
b
∗
(
v
). Let χ be the characteristic function and ν
~
=
ν
χ
D
∖
U
r
(
p
). Then (54)
b
∗
ν
~
=
b
∗
ν
=
b
V
^
.
Since(55)
∫
D
ν
ϕ
n
d
x
d
y
=
∫
D
ν
~
ϕ
n
d
x
d
y
+
∫
U
r
p
ν
ϕ
n
d
x
d
y
,we get (56)
l
i
m
n
→
∞
∫
D
ν
~
ϕ
n
d
x
d
y
=
l
i
m
n
→
∞
∫
D
ν
ϕ
n
d
x
d
y
,due to (52). It follows from (18) and (54) that (57)
l
i
m
n
→
∞
∫
D
ν
~
ϕ
n
d
x
d
y
=
l
i
m
n
→
∞
∫
D
ν
ϕ
n
d
x
d
y
=
b
V
^
=
b
∗
ν
~
.So ν
~ is an infinitesimal asymptotically extremal Beltrami differential in its equivalence class [
[
ν
~
]
]
z and b
(
[
[
ν
~
]
]
z
)
=
b
(
V
^
).
Define(58)
μ
~
z
=
ν
~
z
,
z
∈
D
∖
U
r
p
;
μ
z
,
z
∈
U
r
p
.It is not hard to verify that μ
~
(
z
) is an infinitesimal asymptotically extremal representative of V
^, and by (48) and (49), b
p
∗
(
μ
~
)
<
b
p
(
V
^
)
+
ϵ. This completes the proof of Lemma 2.Theorem 3.
For everyV
^
∈
A
Z
(
D
), if V
^ is a nonsubstantial point, that is, b
p
(
V
^
)
<
b
(
V
^
) for some point p
∈
∂
D, then there exist infinitely many geodesic segments connecting V
^ and the basepoint in A
Z
(
D
).Proof.
Letϵ
=
b
(
V
^
)
-
b
p
(
V
^
). By Lemma 2, there exists an infinitesimal asymptotic extremal representative μ of V
^ such that (59)
b
p
∗
μ
<
b
p
V
^
+
ϵ
.So there exists a positive number r
0 and a neighbourhood U
r
0
=
{
z
∈
D
:
z
-
p
<
r
0
} of p such that (60)
μ
U
r
0
∞
<
b
V
^
.
Setb
(
V
^
)
=
k, μ
|
U
r
0
∞
=
k
′ and δ
=
1
-
k
′
/
k. Since μ is infinitesimally asymptotically extremal, b
∗
(
μ
)
=
b
(
V
)
=
k. From (18), there exists an asymptotic Hamilton sequence {
ψ
n
}
⊂
Q
d
(
R
) such that (61)
l
i
m
n
→
∞
∫
R
μ
ψ
n
=
b
∗
μ
.Furthermore, by Lemma 1, we have (62)
l
i
m
n
→
∞
∫
U
r
0
ψ
n
=
0
,for sufficiently small r
0.
Letχ
r
0 be the characteristic function of U
r
0 and let ϕ
n be the special sequence degenerating towards p constructed as above. For every 0
≤
ρ
≤
δ and t
∈
[
0
,
k
], let (63)
γ
ρ
t
=
t
μ
k
+
ρ
/
2
t
k
-
t
χ
r
0
ϕ
ϕ
z
,where ϕ
=
∑
n
=
1
∞
‍
ϕ
n is a holomorphic quadratic differential on D. Clearly, γ
ρ
(
0
)
=
[
[
0
]
]
z and γ
ρ
(
k
)
=
[
[
μ
]
]
z.
Now we show thatγ
ρ
(
[
0
,
k
]
) is a geodesic segment in A
Z
(
D
). Let 0
≤
t
1
≤
t
2
≤
k. We discuss the infinitesimally asymptotic equivalence class (64)
t
2
-
t
1
μ
k
+
ρ
/
2
k
-
t
1
-
t
2
χ
r
0
ϕ
ϕ
z
.By Lemma 1, it is easy to calculate that (65)
l
i
m
n
→
∞
∫
D
t
2
-
t
1
μ
k
+
ρ
/
2
k
-
t
1
-
t
2
χ
r
ϕ
ϕ
ψ
n
=
t
2
-
t
1
.Since (66)
b
∗
t
2
-
t
1
μ
k
+
ρ
/
2
k
-
t
1
-
t
2
χ
r
0
ϕ
ϕ
=
t
2
-
t
1
,we obtain that (67)
t
2
-
t
1
μ
k
+
ρ
/
2
k
-
t
1
-
t
2
χ
r
0
ϕ
ϕis infinitesimally asymptotically extremal in its equivalence class. From (16), we have (68)
d
¯
γ
ρ
t
1
,
γ
ρ
t
2
=
b
t
2
-
t
1
μ
k
+
ρ
/
2
k
-
t
1
-
t
2
χ
r
0
ϕ
ϕ
z
=
t
2
-
t
1
,which implies that γ
ρ
(
[
0
,
k
]
) is a geodesic segment joining [
[
0
]
]
z and [
[
μ
]
]
z in A
Z
(
D
).
Now we prove that, for0
≤
ρ
1
<
ρ
2
≤
δ, γ
ρ
1
(
[
0
,
k
]
) and γ
ρ
2
(
[
0
,
k
]
) are two different geodesics joining [
[
0
]
]
z and [
[
μ
]
]
z in A
Z
(
D
). Otherwise, suppose γ
ρ
1
(
t
)
=
γ
ρ
2
(
t
), t
∈
(
0
,
k
) when ρ
1
<
ρ
2. Then (69)
ρ
2
-
ρ
1
/
2
t
k
-
t
χ
r
0
ϕ
ϕ
z
∈
Z
0
D
.From (15), for the special sequence ϕ
n degenerating towards p constructed as above, it yields (70)
l
i
m
n
→
∞
∫
D
χ
r
0
ϕ
ϕ
ϕ
n
=
0
.
On the other hand, there exists a positive number sequence{
r
n
} corresponding to {
ϕ
n
} with r
n
<
r
n
-
1
<
⋯
<
r
0 and l
i
m
n
→
∞
r
n
=
0 such that (71)
∫
D
χ
r
0
ϕ
ϕ
ϕ
n
=
∫
U
r
0
∖
U
r
n
-
1
ϕ
ϕ
ϕ
n
+
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
n
+
∫
U
r
n
ϕ
ϕ
ϕ
n
.It follows from (38) that (72)
l
i
m
n
→
∞
∫
U
r
n
ϕ
ϕ
ϕ
n
≤
l
i
m
n
→
∞
∫
U
r
n
ϕ
n
=
0
,
l
i
m
n
→
∞
∫
U
r
0
∖
U
r
n
-
1
ϕ
ϕ
ϕ
n
≤
l
i
m
n
→
∞
∫
U
r
0
∖
U
r
n
-
1
ϕ
n
=
0
.So (73)
l
i
m
n
→
∞
∫
U
r
n
ϕ
ϕ
ϕ
n
=
0
,
l
i
m
n
→
∞
∫
U
r
0
∖
U
r
n
-
1
ϕ
ϕ
ϕ
n
=
0
.
Furthermore,(74)
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
n
=
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
+
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
n
-
ϕ
.It follows from (41) and (42) that (75)
l
i
m
n
→
∞
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
n
-
ϕ
=
0
,
l
i
m
n
→
∞
∫
U
r
n
-
1
∖
U
r
n
ϕ
=
1
.Therefore, (76)
l
i
m
n
→
∞
∫
U
r
n
-
1
∖
U
r
n
ϕ
ϕ
ϕ
n
=
1
.By (71)–(76), we conclude that (77)
l
i
m
n
→
∞
∫
D
χ
r
0
ϕ
ϕ
ϕ
n
=
1
.This is a contradiction with (70), which implies γ
ρ
1
(
t
)
≠
γ
ρ
2
(
t
) for t
∈
(
0
,
k
) if ρ
1
≠
ρ
2.
Thus, we have constructed infinitely many geodesicsγ
ρ
(
t
) (0
≤
ρ
≤
δ) joining [
[
0
]
] and [
[
μ
]
] in A
Z
(
D
).The situation on the geodesics joining an infinitesimal substantial point with the basepoint is not clear. We conjecture that there exist infinitely many geodesics between an infinitesimal substantial point and the basepoint inA
Z
(
D
).



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*Source: 276719-2015-12-20.xml*