5.1 Biholomorphy
In Section 2.4, it was asserted that for a holomorphic function , the map is conformal when .
We have the following immediate assertion:
Theorem 5.1.1 (Open Mapping Theorem).
Suppose is a region (open, nonempty, and connected). Then the image of any holomorphic and non-constant function , , is a region.
Proof.
The nonemptiness of is an immediate conclusion from the fact that is nonempty and is defined on all of .
Let be an arbitrary point in . Then such that . Since is non-constant, the function has an isolated zero at . Thus for sufficiently small , the only zero of in is at .
By Theorem 3.3.7, then there exists such that , has exactly one zero in . In other words, , such that , such that . Thus, . Thus, is an open set since each contained point has a fully contained open neighborhood.
Let be arbitrary and distinct. Then there exist such that and . By the connectivity of , there exists a path that connects and . Then is a curve that joins and . Thus, is connected.
Holomorphic injectivity, or univalence, satisfies the proceeding assertion:
Lemma 5.1.1.
Let be a region and suppose is univalent. Then is non-vanishing on .
Proof.
Suppose, for the sake of contradiction, that is univalent on such that such that . Let . The previous statement is equivalent to: has a zero at with multiplicity .
Since is univalent, neither nor may have accumulation points in . Thus, such that is the only zero of either and contained in . By Theorem 3.3.7, such that , the equation has solutions in , which cannot lie all at a single point (unless that point is itself, which cannot be the case as already maps to ), as otherwise would not be the only zero of in . This contradicts the univalence of .
Conversely, we have the following statement on local univalence and invertibility.
Theorem 5.1.2.
Let be a region and suppose is holomorphic. If for some , then there exists an open neighborhood of on which is univalent.
Proof.
Let . Since and , it follows that is a simple zero of . Let be an open neighborhood (relatively compact in ) of whose closure does not contain other zeros of . By Theorem 3.3.7, such that , has only one solution for satisfying . Therefore, we can choose a relatively compact open subset of such that , on which is univalent.
Moreover, if is univalent and surjective, mapping to , then its inverse is univalent on . Such bijective holomorphic functions are known as biholomorphisms or biholomorphic functions.
We will now study holomorphic functions from a more geometric perspective.
Theorem 5.1.3.
Let be a region, and let be a rectifiable simple closed counterclockwise-oriented curve that is null-homotopic in . Denote by . If is holomorphic and maps injectively to a simple closed curve , then is univalent in , , and is traversed counterclockwise.
Proof.
Let . Let be the number of zeros of in . By the Argument Principle (Theorem 3.3.4), for ,
- If , the expression vanishes since . Then has no solution in (i.e. ).
- If , then winds around exactly once, and hence, in other words, , has a unique solution in (i.e. ).
-
If lies on , then it can be shown that has no zeros in .
Indeed, for the sake of contradiction, assume that such that . By the Open Mapping Theorem (Theorem 5.1.1), such that , or equivalently, , has zeros in . Since lies on , a subset of lies in the exterior of . It was previously established that has no zeros if . Thus, we have a contradiction, and no such exists, implying .
We then have
Hence, is univalent in (since for each , is at most one).
Moreover, any point must map to either or . The latter is an impossibility since otherwise . This .
We will now give examples of biholomorphisms.
Example 5.1.1.
The only biholomorphisms which map to itself are in the form of
This follows directly from Theorem 3.5.1.
Example 5.1.2.
The only biholomorphisms which map to are in the form of
Proof.
First assume . It follows that
Therefore, this transformation maps to . The inverse mapping is equal to
Assume . We then have
Hence, maps to univalently and surjectively since it is also an element in .
Let be the biholomorphism from to in the form of (for and , known as the Cayley transform). Let be an arbitrary biholomorphism from to . It follows that is a holomorphic automorphism on . Since , we have
Obviously, attains every value on the unit disk for varying and . Similarly, the values attained by cover the upper half-plane for (since it is in the form of (5.1.3)). Thus, all biholomorphisms from to are in the form of (5.1.2).
Let us now introduce some important properties of linear fractional transformations. By Proposition 4.4.1, it follows that the composition of two linear fractional transformations is also a linear fractional transformation.
Theorem 5.1.4.
Let be the collection of subsets of that are circles or , where is a straight line in (known as generalized circles). Then every linear fractional transformation maps elements of to elements of .
Proof.
Since each linear fractional transformation is a composition of maps in the form of , , and , it suffices to show that these maps preserve the property of being a circle or a straight line. Consider a circle defined implicitly with
For , this can be rewritten as
If , the equation represents a straight line. It is easy to see that a complex dilation or a translation of will preserve the property of being a straight line or a circle. Indeed, by letting for nonzero in (5.1.4), we have
which is trivially in the form of (5.1.4). Similarly, if we substitute , we have
If we substitute , we have
which is in the form of (5.1.4).
Remark.
As in Example 5.1.2, we can consider extended straight lines in the form of as generalized circles in the Riemann sphere. In other words, the extended line can be geometrically visualized by a circle with infinite radius. In fact, when a circle on the Riemann sphere is projected stereographically onto the complex plane, the result is always either a circle or a straight line.
Definition 5.1.1 (Cross-Ratio).
Let be points such that at least three of them are distinct. The cross-ratio of these points is defined as
If at least one of the four points is , then the cross-ratio is defined by the limit:
One important property of the cross-ratio is that it is invariant under linear fractional transformations. In other words, if is a linear fractional transformation, then
The proof is trivial and can be verified by substituting the definition of the linear fractional transformation into the definition of the cross-ratio.
Furthermore, if a function is invariant under the group of linear fractional transformations, then it is a function of the cross-ratio. In other words, the cross-ratio is the only invariant under the group of linear fractional transformations . Indeed, suppose that
We aim to show that is a function of a cross-ratio. Let
be a linear fractional transformation. Then we have
which is a function of the cross-ratio.