# Some Surfaces with Zero Curvature inℍ

Authors: Dae Won Yoon
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We study surfaces defined as graph of the functionz
=
f
(
x
,
y
) in the product space ℍ
2
×
ℝ. In particular, we completely classify flat or minimal surfaces given by f
(
x
,
y
)
=
u
(
x
)
+
v
(
y
), where u
(
x
) and v
(
y
) are smooth functions.


---

## Body

## 1. Introduction

Homogenous geometries have main roles in the modern theory of manifolds. Homogenous spaces are, in a sense, the nicest examples of Riemannian manifolds and have applications in physics [1]. To underline their importance from the mathematical point of view we roughly cite the famous Thurston conjecture. This conjecture asserts that every compact orientable 3-dimensional manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure from among the eight maximal simple connected homogenous Riemannian 3-dimensional geometries [2]. The Riemannian product space ℍ
2
×
ℝ is one of the eight model spaces.Constant mean curvature and constant Gaussian curvature surfaces are one of the main objects which have drawn geometers' interest for a very long time. Recently, the study of the geometry of surfaces inℍ
2
×
ℝ is growing very rapidly, and the interest is mainly focused on minimal and constant mean curvature surfaces [3–9].The purpose of this paper is to study surfaces defined as graph of the functionz
=
f
(
x
,
y
) in the product space ℍ
2
×
ℝ. In Sections 4 and 5 we classify minimal and flat surfaces defined as f
(
x
,
y
)
=
u
(
x
)
+
v
(
y
), where u
(
x
) and v
(
y
) are smooth functions.


## 2. Preliminaries

Letℍ
2
=
{
(
x
,
y
)
∈
ℝ
2
∣
y
>
0
} be the upper half plane model of the hyperbolic plane endowed with the metric, of constant Gaussian curvature -
1, given by
(1)
g
ℍ
=
(
d
x
2
+
d
y
2
)
y
2
.
The hyperbolic space ℍ
2, with the group structure derived by the composition of proper affine maps, is a Lie group and the metric g
ℍ is left invariant. Therefore, the product space ℍ
2
×
ℝ is a Lie group with the left invariant product metric
(2)
g
=
d
x
2
+
d
y
2
y
2
+
d
z
2
.
On the other hand, an orthonormal basis of left invariant vector fields on ℍ
2
×
ℝ is
(3)
E
1
=
y
∂
∂
x
,
E
2
=
y
∂
∂
y
,
E
3
=
∂
∂
z
with the only nontrivial commutator relation [
E
1
,
E
2
]
=
-
E
1. It follows that the Levi-Civita connection ∇
~ of ℍ
2
×
ℝ is expressed as
(4)
∇
~
E
1
E
1
=
E
2
,
∇
~
E
1
E
2
=
-
E
1
,
∇
~
E
1
E
3
=
0
,
∇
~
E
2
E
1
=
0
,
∇
~
E
2
E
2
=
0
,
∇
~
E
2
E
3
=
0
,
∇
~
E
3
E
1
=
0
,
∇
~
E
3
E
2
=
0
,
∇
~
E
3
E
3
=
0
.
For any vectors X
=
x
1
E
1
+
y
1
E
2
+
z
1
E
3 and Y
=
x
2
E
1
+
y
2
E
2
+
z
2
E
3 in ℍ
2
×
ℝ the cross-product × is defined by
(5)
X
×
Y
=
(
y
1
z
2
-
y
2
z
1
)
E
1
+
(
x
2
z
1
-
x
1
z
2
)
E
2
+
(
x
1
y
2
-
x
2
y
1
)
E
3
.


## 3. Graphs inℍ
2
×
ℝ

Let us consider a surfaceΣ parametrized by
(6)
ϕ
(
x
,
y
)
=
(
x
,
y
,
f
(
x
,
y
)
)
,
(
x
,
y
)
∈
Ω
,
where Ω is a domain in ℍ
2 and f
:
Ω
→
ℝ is a smooth function. Then Σ is a surface defined as graph of the function f defined on Ω
⊂
ℍ
2. In this case, we have
(7)
e
1
:
=
ϕ
x
=
(
1,0
,
f
x
)
=
1
y
E
1
+
f
x
E
3
,
e
2
:
=
ϕ
y
=
(
0,1
,
f
y
)
=
1
y
E
2
+
f
y
E
3
.
It follows that the coefficients of the first fundamental form of Σ are given by
(8)
E
=
g
(
ϕ
x
,
ϕ
x
)
=
f
x
2
+
1
y
2
,
F
=
g
(
ϕ
x
,
ϕ
y
)
=
f
x
f
y
,
G
=
g
(
ϕ
y
,
ϕ
y
)
=
f
y
2
+
1
y
2
.
Also, the unit normal vector field U to Σ is given by
(9)
U
(
x
,
y
)
=
-
f
x
ω
y
E
1
-
f
y
ω
y
E
2
+
1
ω
y
2
E
3
,
where
(10)
ω
=
1
y
2
y
2
(
f
x
2
+
f
y
2
)
+
1
.
By a straightforward calculation, we obtain
(11)
∇
~
e
1
e
1
=
1
y
2
E
2
+
f
x
x
E
3
,
∇
~
e
1
e
2
=
-
1
y
2
E
1
+
f
x
x
E
3
,
∇
~
e
2
e
2
=
-
1
y
2
E
2
+
f
y
y
E
3
,
which imply that the coefficients of the second fundamental form of Σ are
(12)
L
=
g
(
∇
~
e
1
e
1
,
U
)
=
y
f
x
x
-
f
y
ω
y
3
,
M
=
g
(
∇
~
e
1
e
2
,
U
)
=
y
f
x
y
+
f
x
ω
y
3
,
N
=
g
(
∇
~
e
2
e
2
,
U
)
=
y
f
y
y
+
f
y
ω
y
3
.
Thus, from (8) and (12) the Gaussian curvature K and the mean curvature H are, respectively,
(13)
K
=
1
ω
4
y
6
(
(
y
f
x
x
-
f
y
)
(
y
f
y
y
+
f
y
)
-
(
y
f
x
y
+
f
x
)
2
)
,
H
=
1
2
ω
3
y
4
(
(
1
+
y
2
f
y
2
)
f
x
x
-
y
(
f
x
2
+
f
y
2
)
f
y
=
1
2
ω
3
y
4
2
-
2
y
2
f
x
f
y
f
x
y
+
(
1
+
y
2
f
x
2
)
f
y
y
)
.Proposition 1.
LetΣ be a surface defined as graph of the function f
:
Ω
⊂
ℍ
2
→
ℝ. Then Σ is a minimal surface if and only if
(14)
(
1
+
y
2
f
y
2
)
f
x
x
-
y
(
f
x
2
+
f
y
2
)
f
y
-
2
y
2
f
x
f
y
f
x
y
+
(
1
+
y
2
f
x
2
)
f
y
y
=
0
.Proposition 2.
LetΣ be a surface defined as graph of the function f
:
Ω
⊂
ℍ
2
→
ℝ. Then Σ is flat if and only if
(15)
(
y
f
x
x
-
f
y
)
(
y
f
y
y
+
f
y
)
-
(
y
f
x
y
+
f
x
)
2
=
0
.Remark 3.
Some examples are satisfying the ODE (14) studied in [7]. Also, examples in Lorentz product space ℍ
2
×
ℝ
1 can be found in [10].


## 4. Minimal Surfaces Defined byf
(
x
,
y
)
=
u
(
x
)
+
v
(
y
)

LetΣ be a surface in ℍ
2
×
ℝ parametrized by
(16)
ϕ
(
x
,
y
)
=
(
x
,
y
,
u
(
x
)
+
v
(
y
)
)
for all y
>
0, where u
(
x
) and v
(
y
) are smooth functions. We suppose that Σ is a minimal surface. Then, from (14) we have the following minimal surface equation:
(17)
(
1
+
y
2
(
v
′
)
2
)
u
′′
-
y
(
(
u
′
)
2
+
(
v
′
)
2
)
v
′
+
(
1
+
y
2
(
u
′
)
2
)
v
′′
=
0
.
In order to solve it, divide first by 1
+
y
2
(
v
′
)
2
≠
0; then we get
(18)
u
′′
-
y
(
(
u
′
)
2
+
(
v
′
)
2
)
1
+
y
2
(
v
′
)
2
v
′
+
1
+
y
2
(
u
′
)
2
1
+
y
2
(
v
′
)
2
v
′′
=
0
,
for all x
,
y
∈
Ω. Differentiating with respect to x, we obtain
(19)
u
′′′
+
2
(
y
2
v
′′
-
y
v
′
1
+
y
2
(
v
′
)
2
)
u
′
u
′′
=
0
.
First of all, we suppose that u
′′
=
0 on an open interval; that is, u
(
x
)
=
a
x
+
b
,
a
,
b
∈
ℝ. In this case, from (17) we obtain
(20)
v
′′
-
a
2
y
1
+
a
2
y
2
v
′
-
y
1
+
a
2
y
2
(
v
′
)
3
=
0
.
We put v
′
(
y
)
=
p
(
y
). Then the last equation can be written as
(21)
p
′
-
y
1
+
a
2
y
2
(
a
2
p
+
p
3
)
=
0
.
Its general solution is given by
(22)
p
=
±
c
1
a
1
+
a
2
y
2
1
-
c
1
2
(
1
+
a
2
y
2
)
.
From this, we thus have
(23)
v
(
y
)
=
±
∫
‍
c
1
a
1
+
a
2
y
2
1
-
c
1
2
(
1
+
a
2
y
2
)
d
y
,
where c
1
∈
ℝ.Now, we assume thatu
′′
≠
0 on an open interval, and divide (19) by u
′
u
′′. It follows that
(24)
u
′′′
u
′
u
′′
+
2
y
2
v
′′
-
y
v
′
1
+
y
2
(
v
′
)
2
=
0
.
Hence we deduce the existence of a real number k
∈
ℝ such that
(25)
u
′′′
=
2
k
u
′
u
′′
,
y
2
v
′′
-
y
v
′
=
-
k
(
1
+
y
2
(
v
′
)
2
)
.
Let us distinguish the following cases according to k.Case 1.
Ifk
=
0, then u
′′′
=
0 and y
v
′′
-
v
′
=
0. It follows that u
(
x
)
=
a
1
x
2
+
b
1
x
+
c
1
(
a
1
≠
0
,
b
1
,
c
1
∈
ℝ
). If v
′
=
0, then v
(
y
)
=
a
2
(
a
2
∈
ℝ
). In this case, from (17) we obtain a
1
=
0; it is a contradiction. If v
′
≠
0, then we get v
(
y
)
=
(
1
/
2
)
b
2
y
2
+
c
2
(
b
2
≠
0
,
c
2
∈
ℝ
). In such case, (17) is polynomial equation on x and y. From the coefficients of y
4 and the constant term we have 2
a
1
-
b
2
=
0 and 2
a
1
+
b
2
=
0, which imply a
1
=
0 and b
2
=
0. It is a contradiction.Case 2.
Ifk
≠
0, then from the first equation in (25) we have
(26)
u
′′
=
e
2
k
u
+
d
1
,
where d
1
∈
ℝ. Let
(27)
u
=
1
2
k
(
-
d
1
+
ln
⁡
g
)
be any solution of (26), where g is a smooth function. Then (26) can be rewritten as
(28)
g
g
′′
-
(
g
′
)
2
=
2
k
g
3
.
We put p
=
g
′. Then, we have
(29)
d
p
d
g
-
1
g
p
=
2
k
g
2
p
-
1
.
We again put t
=
p
2. In this case the above equation becomes
(30)
d
t
d
g
-
2
g
t
=
4
k
g
2
and its general solution is given by
(31)
t
=
g
2
(
4
k
g
+
c
1
)
.
Thus, we get
(32)
d
g
d
x
=
±
g
4
k
g
+
c
1
.
After an integration, we can find
(33)
g
=
c
1
4
k
tan
2
(
8
k
2
c
1
(
±
x
+
c
2
)
)
-
c
1
4
k
,
where c
2
∈
ℝ. By combining (27) and (33), we thus have
(34)
u
(
x
)
=
1
2
k
[
-
d
1
+
ln
⁡
(
c
1
4
k
tan
2
(
8
k
2
c
1
(
±
x
+
c
2
)
)
-
c
1
4
k
)
]
.Now, we consider the second equation in (25). Since y
>
0, we yield
(35)
v
′′
+
k
y
2
-
1
y
v
′
+
k
(
v
′
)
2
=
0
.
We put p
=
v
′. Then, the above equation becomes
(36)
p
′
+
k
y
2
-
1
y
p
+
k
p
2
=
0
.
Since k
≠
0, without loss of generality we take k
=
1 or k
=
-
1.Subcase i. Let k
=
1. We do the change
(37)
p
=
1
y
+
1
h
(
y
)
,
where h is a nonzero smooth function. Then, (36) can be rewritten as the form
(38)
h
′
-
1
y
h
=
1
.
Thus, its general solution is
(39)
h
(
y
)
=
y
(
ln
⁡
y
+
c
1
)
,
where c
1
∈
ℝ. So, p
=
(
1
/
y
)
+
(
1
/
y
(
ln
⁡
y
+
c
1
)
) and from its integration we can obtain
(40)
v
(
y
)
=
ln
⁡
(
c
2
y
ln
⁡
(
y
+
c
1
)
)
,
where c
2
∈
ℝ.Subcase ii. Let k
=
-
1. We put
(41)
p
=
-
1
y
+
1
h
(
y
)
,
where h is a nonzero smooth function. Then, (36) becomes
(42)
h
′
-
1
y
h
=
-
1
and its general solution is given by
(43)
h
(
y
)
=
-
y
(
ln
⁡
y
+
c
1
)
,
where c
1
∈
ℝ. Thus, we have
(44)
v
(
y
)
=
-
ln
⁡
(
c
2
y
ln
⁡
(
y
+
c
1
)
)
,
where c
2
∈
ℝ. The surface given by (34) and (44) is shown in Figure 1.Figure 1
A minimal surface defined by (34) and (44).Consequently, we have the following.Theorem 4.
LetΣ be a surface defined as graph of the function f
(
x
,
y
)
=
u
(
x
)
+
v
(
y
). If Σ is a minimal surface, then Σ is parametrized as
(45)
ϕ
(
x
,
y
)
=
(
x
,
y
,
u
(
x
)
+
v
(
y
)
)
,
where(1)
u
(
x
)
=
a
x
+
b and v
(
y
)
=
±
∫
‍
(
c
1
a
1
+
a
2
y
2
/
1
-
c
1
2
(
1
+
a
2
y
2
)
)
d
y with a
,
b
,
c
1
∈
ℝ, or
(2)
u
(
x
)
=
(
1
/
2
k
)
[
-
c
3
+
ln
⁡
(
(
c
1
/
4
k
)
tan
2
(
8
k
2
c
1
(
±
x
+
c
2
)
)
-
(
c
1
/
4
k
)
)
] and v
(
y
)
=
±
ln
⁡
(
d
1
y
ln
⁡
(
y
+
d
2
)
) with k
≠
0
,
c
1
,
c
2
,
c
3
,
d
1
,
d
2
∈
ℝ.


## 5. Flat Surfaces Defined byf
(
x
,
y
)
=
u
(
x
)
+
v
(
y
)

LetΣ be a surface defined by (16). Assume that Σ is a flat surface. Then, from (15) we have the following flat surface equation:
(46)
y
(
y
v
′′
+
v
′
)
u
′′
-
(
y
v
′′
+
v
′
)
v
′
-
(
u
′
)
2
=
0
.
In order to solve it, differentiating with respect to x, we have
(47)
y
(
y
v
′′
+
v
′
)
d
d
x
(
u
′′
)
-
d
d
x
(
(
u
′
)
2
)
=
0
.
Thus, there exists a nonzero real number k such that
(48)
d
d
x
(
u
′′
)
=
k
d
d
x
(
(
u
′
)
2
)
,
y
(
y
v
′′
+
v
′
)
=
1
k
.
From the first equation in (48), we get
(49)
u
′′
=
k
(
u
′
)
2
+
c
1
,
where c
1
∈
ℝ. We put p
=
u
′, and it follows that we yield
(50)
d
p
d
u
=
k
p
2
+
c
1
p
.
From this, the general solution is
(51)
p
=
±
1
k
e
2
k
(
u
+
c
2
)
-
c
1
k
,
where c
2
∈
ℝ. We can assume that c
1
=
0. From the last equation we can easily obtain (see Figure 2)
(52)
u
(
x
)
=
±
1
k
(
ln
⁡
(
-
k
(
x
+
c
3
)
)
+
k
c
2
)
,
where c
3
∈
ℝ.Figure 2
A flat surface defined by (52) and (55).In order to solve the second equation in (48), divide by y
2 and put q
=
v
′. Then, we get
(53)
q
′
+
1
y
q
=
1
k
y
2
and its general solution is given by
(54)
q
=
1
y
(
1
k
ln
⁡
y
+
d
1
)
,
where d
1
∈
ℝ. From this, we thus obtain (see Figure 2)
(55)
v
(
y
)
=
1
2
k
(
ln
⁡
y
)
2
+
d
1
ln
⁡
y
+
d
2
,
where d
2
∈
ℝ.As a conclusion, we have the following.Theorem 5.
LetΣ be a surface defined as graph of the function f
(
x
,
y
)
=
u
(
x
)
+
v
(
y
). If Σ is a flat surface, then Σ is parametrized as
(56)
ϕ
(
x
,
y
)
=
(
x
,
y
,
u
(
x
)
+
v
(
y
)
)
,
where u
(
x
)
=
±
(
1
/
k
)
(
ln
⁡
(
-
k
(
x
+
c
1
)
)
+
k
c
2
) and v
(
y
)
=
(
1
/
2
k
)
(
ln
⁡
y
)
2
+
d
1
ln
⁡
y
+
d
1 with k
≠
0
,
c
1
,
c
2
,
d
1
,
d
2
∈
ℝ.



---

*Source: 154294-2014-03-24.xml*