Local uniformizing parameter

local uniformizer, local parameter

A complex variable $ t $ defined as a continuous function $ t _ {p _ {0} } = \phi _ {p _ {0} } ( p) $ of a point $ p $ on a Riemann surface $ R $, defined everywhere in some neighbourhood $ V ( p _ {0} ) $ of a point $ p _ {0} \in R $ and realizing a homeomorphic mapping of $ V ( p _ {0} ) $ onto the disc $ D ( p _ {0} ) = \{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) } \} $, where $ \phi _ {p _ {0} } ( p _ {0} ) = 0 $. Here $ V ( p _ {0} ) $ is said to be a distinguished or parametric neighbourhood, $ \phi _ {p _ {0} } : V ( p _ {0} ) \rightarrow D ( p _ {0} ) $ a distinguished or parametric mapping, and $ D ( p _ {0} ) $ a distinguished or parametric disc. Under a parametric mapping any point function $ g ( p) $, defined in a parametric neighbourhood $ V ( p _ {0} ) $, goes into a function of the local uniformizing parameter $ t $, that is, $ g ( p) = g [ \phi _ {p _ {0} } ^ {-} 1 ( t) ] = G ( t) $. If $ V ( p _ {0} ) $ and $ V ( p _ {1} ) $ are two parametric neighbourhoods such that $ V ( p _ {0} ) \cap V ( p _ {1} ) \neq \emptyset $, and $ t _ {p _ {0} } $ and $ t _ {p _ {1} } $ are the two corresponding local uniformizing parameters, then $ t _ {p _ {1} } = \phi _ {p _ {1} } [ \phi _ {p _ {0} } ^ {-} 1 ( t _ {p _ {0} } )] $ is a univalent holomorphic function on some subdomain of $ D ( p _ {0} ) $ realizing a biholomorphic mapping of this subdomain into $ D ( p _ {1} ) $.

If $ R = R _ {F} $ is the Riemann surface of an analytic function $ w = F ( z) $ and $ p _ {0} $ is a regular element of $ F ( z) $ with projection $ z _ {0} \neq \infty $, then $ t _ {p _ {0} } = z - z _ {0} $; $ t _ {p _ {0} } = 1 / z $ for $ z _ {0} = \infty $. If $ p _ {0} $ is a singular, or algebraic, element of $ F ( z) $, corresponding to the branch point $ z _ {0} $ of order $ k - 1 > 0 $, then $ t _ {p _ {0} } = ( z - z _ {0} ) ^ {1/k} $ for $ z _ {0} \neq \infty $ and $ t _ {p _ {0} } = 1 / z ^ {1/k} $ for $ z _ {0} = \infty $. In a parametric neighbourhood of an element $ p _ {0} $ the local uniformizing parameter $ t $ actually realizes a local uniformization, generally speaking, of the many-valued relation $ w = F ( z) $, according to the formulas (for example, for $ z _ {0} \neq \infty $):

$$ z = z _ {0} + t ^ {k} ,\ \ w = F ( z _ {0} + t ^ {k} ) = w ( t) ,\ \ k \geq 1 . $$

In the case when $ R $ is a Riemann surface with boundary, for points $ p _ {0} $ belonging to the boundary of $ R $ the local uniformizing parameter $ t _ {p _ {0} } = \phi _ {p _ {0} } ( p) $ maps the parametric neighbourhood $ V ( p _ {0} ) $ onto the half-disc

$$ D ( p _ {0} ) = \ \{ {t \in \mathbf C } : {| t | < r ( p _ {0} ) , \mathop{\rm Im} t \geq 0 } \} . $$

If $ R $ is a a Riemannian domain over a complex space $ \mathbf C ^ {n} $, $ n > 1 $, then the local uniformizing parameter

$$ t _ {p _ {0} } = \ \phi _ {p _ {0} } ( p) = \ ( t _ {1} \dots t _ {n} ) _ {p _ {0} } = \ ( \phi _ {1} ( p) \dots \phi _ {n} ( p) ) _ {p _ {0} } $$

realizes a homeomorphic mapping of the parametric neighbourhood $ V ( p _ {0} ) $ onto the polydisc

$$ D ( p _ {0} ) = $$

$$ = \ \{ t = ( t _ {1} \dots t _ {n} ) \in \mathbf C ^ {n} : | t _ {1} | < r _ {1} ( p _ {0} ) \dots | t _ {n} | < r _ {n} ( p _ {0} ) \} . $$

If $ V ( p _ {0} ) \cap V ( p _ {1} ) $ is not empty, then the mapping $ t _ {p _ {1} } = \phi _ {p _ {1} } [ \phi _ {p _ {0} } ^ {-} 1 ( t _ {p _ {0} } ) ] $ biholomorphically maps a certain subdomain of $ D ( p _ {0} ) $ into $ D ( p _ {1} ) $.

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)

References

[a1]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980)