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)