In the
mathematical theory
A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area o ...
of functions of
one or
more complex variables, and also in
complex algebraic geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, a biholomorphism or biholomorphic function is a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
whose
inverse is also
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
.
Formal definition
Formally, a ''biholomorphic function'' is a function
defined on an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
''U'' of the
-dimensional complex space C
''n'' with values in C
''n'' which is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
and
one-to-one, such that its
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
is an open set
in C
''n'' and the inverse
is also
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
. More generally, ''U'' and ''V'' can be
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
s. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11).
If there exists a biholomorphism
, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorphic.
Riemann mapping theorem and generalizations
If
every
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
open set other than the whole complex plane is biholomorphic to the
unit disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
(this is the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...
). The situation is very different in higher dimensions. For example, open
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
s and open unit
polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.
More specifically, if we denote by D(z,r) the open disc of center ''z'' and radius ''r'' in the complex plane, then a ...
s are not biholomorphically equivalent for
In fact, there does not exist even a
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
holomorphic function from one to the other.
Alternative definitions
In the case of maps ''f'' : ''U'' → C defined on an open subset ''U'' of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a
conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
to be an injective map with nonzero derivative i.e., ''f''’(''z'')≠ 0 for every ''z'' in ''U''. According to this definition, a map ''f'' : ''U'' → C is conformal if and only if ''f'': ''U'' → ''f''(''U'') is biholomorphic. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, without requiring that the map be injective. According to this weaker definition of conformality, a conformal map need not be biholomorphic even though it is locally biholomorphic. For example, if ''f'': ''U'' → ''U'' is defined by ''f''(''z'') = ''z''
2 with ''U'' = C–, then ''f'' is conformal on ''U'', since its derivative ''f''’(''z'') = 2''z'' ≠ 0, but it is not biholomorphic, since it is 2-1.
References
*
*
*
*
*
{{PlanetMath attribution, urlname=BiholomorphicallyEquivalent, title=biholomorphically equivalent
Several complex variables
Algebraic geometry
Complex manifolds
Functions and mappings