Local uniformizing parameter
local uniformizer, local parameter
A complex variable 
defined as a continuous function 
of a point 
on a Riemann surface 
, defined everywhere in some neighbourhood 
of a point 
and realizing a homeomorphic mapping of 
onto the disc 
, where 
. Here 
is said to be a distinguished or parametric neighbourhood, 
a distinguished or parametric mapping, and 
a distinguished or parametric disc. Under a parametric mapping any point function 
, defined in a parametric neighbourhood 
, goes into a function of the local uniformizing parameter 
, that is, 
. If 
and 
are two parametric neighbourhoods such that 
, and 
and 
are the two corresponding local uniformizing parameters, then 
is a univalent holomorphic function on some subdomain of 
realizing a biholomorphic mapping of this subdomain into 
.
If 
is the Riemann surface of an analytic function 
and 
is a regular element of 
with projection 
, then 
; 
for 
. If 
is a singular, or algebraic, element of 
, corresponding to the branch point 
of order 
, then 
for 
and 
for 
. In a parametric neighbourhood of an element 
the local uniformizing parameter 
actually realizes a local uniformization, generally speaking, of the many-valued relation 
, according to the formulas (for example, for 
):
 |
In the case when 
is a Riemann surface with boundary, for points 
belonging to the boundary of 
the local uniformizing parameter 
maps the parametric neighbourhood 
onto the half-disc
 |
If 
is a a Riemannian domain over a complex space 
, 
, then the local uniformizing parameter
 |
realizes a homeomorphic mapping of the parametric neighbourhood 
onto the polydisc
 |
 |
If 
is not empty, then the mapping 
biholomorphically maps a certain subdomain of 
into 
.
References
| [1] | A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) |
| [2] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 |
| [3] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) |
Comments
References
| [a1] | H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) |
Local uniformizing parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_uniformizing_parameter&oldid=47685