Affiliation:
1. Department of Mathematics , Kyungpook National University , 41566 Daegu , Republic of Korea
2. Department of Mathematics Education and RINS , Gyeongsang National University , 52828 Jinju , Republic of Korea
Abstract
Abstract
In this paper, we study a conformally flat 3-space
𝔽
3
{\mathbb{F}_{3}}
which is an Euclidean 3-space with a conformally flat metric
with the conformal factor
1
F
2
{\frac{1}{F^{2}}}
, where
F
(
x
)
=
e
-
x
1
2
-
x
2
2
{F(x)=e^{-x_{1}^{2}-x_{2}^{2}}}
for
x
=
(
x
1
,
x
2
,
x
3
)
∈
ℝ
3
{x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}}
.
In particular, we construct all helicoidal surfaces in
𝔽
3
{\mathbb{F}_{3}}
by solving the second-order non-linear ODE with extrinsic curvature and
mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as
examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in
𝔽
3
{\mathbb{F}_{3}}
.
Funder
National Research Foundation of Korea
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