Abstract
AbstractIn this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let $$E\not \subset {\hat{\mathbb {R}}}$$
E
⊄
R
^
be an arc or a circle. If a map $$f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}$$
f
:
C
^
↦
C
^
preserves cross-ratios in E, then f is a Möbius transformation when $${\bar{E}}\not =E$$
E
¯
≠
E
and f is a Möbius or conjugate Möbius transformation when $${\bar{E}}=E$$
E
¯
=
E
, where $${\bar{E}}=\{{\bar{z}}|z\in E\}$$
E
¯
=
{
z
¯
|
z
∈
E
}
. (II) Let $$E\subset {\hat{\mathbb {R}}}$$
E
⊂
R
^
be an arc satisfying the condition that the cardinal number $$\#(E\cap \{0,1,\infty \})<2$$
#
(
E
∩
{
0
,
1
,
∞
}
)
<
2
. If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption $$\#(E\cap \{0,1,\infty \})<2$$
#
(
E
∩
{
0
,
1
,
∞
}
)
<
2
is necessary.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Analysis