# 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)