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Local uniformizing parameter

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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)
How to Cite This Entry:
Local uniformizing parameter. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Local_uniformizing_parameter&oldid=11803
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098