8.3 From Schwarz–Pick to Ahlfors and Value Distribution of Entire Functions
While Schwarz Lemma in Lemma 3.5.1 concerns self-maps of with a fixed point at the origin, the Schwarz–Pick Lemma in Lemma 3.5.2 generalizes this to arbitrary points in as well as the hyperbolic contraction property of holomorphic maps.
In 1938, Lars Ahlfors provided a further generalization by curvature, prompting the study of complex functions from a differential-geometric approach.
The hyperbolic metric in (8.3.2) does not increase under any holomorphic . It was realized that this was a consequence of the constant negative curvature of . The results we now provide are simplifications of those from [2].
Theorem 8.3.1 (Schwarz–Ahlfors–Pick).
Let be holomorphic on . Suppose that is a regular metric defined on an open neighborhood , where , , and for all . Then
where is the Poincaré metric, and equivalently,
or that the metric does not exceed the hyperbolic metric under the map .
Proof.
Define
to generalize the Poincaré metric to . (8.3.5) gives that for any . Define the real-valued function
which is nonnegative and continuous on . The pullback metric is continuous on and thus bounded on (as a consequence of Theorem 1.2.13). As , , and hence . Thus,
must be attained at some (within the interior).
If , then , by maximality. On the contrary, if , has well-defined Gaussian curvature at . Since
By assumption, we have . Hence, . Since is increasing in , is a local maximum of and hence . Thus, we have
Now let , and it follows that .
Theorem 8.3.2.
Let be holomorphic. Let (where ) define a regular metric such that at every point , either
-
The second derivatives of are continuous () and
-
There exist two opposite directions , such that
(the directional derivatives).
Then the metric does not exceed the hyperbolic metric .
Proof.
The first case is equivalent to .
The only modification to the proof of Theorem 8.3.1 is to consider the case of the inequality involving directional derivatives for each (which by definition, is where the maximum value of is attained within ). By the increasing nature of , is also a local maximum of .
Since is a local maximum, we must have
This implies that
by the symmetry of the hyperbolic metric and the fact that the two directions are opposite to each other. Pulling back to contradicts with (8.3.3). Thus, cannot both simultaneously be the location of a maximum while satisfying said inequality; therefore the theorem follows.
Theorem 8.3.3.
Let be holomorphic. Let
be a continuous conformal metric (but not necessarily ) such that at each point , there exists a neighborhood in and a regular metric thereon such that and everywhere else (referred to as a “supporting metric”). If each everywhere, then the conclusion of Theorem 8.3.1 continues to hold for .
Proof.
By assumption, we have
Let , which attains is maximum of at as well.
The calculations in Theorem 8.3.1 on (whose curvature calculations are now valid by ) give that , which implies . (We have used the supporting metric, rather than , to derive this inequality) The rest of the theorem follows naturally.
Theorem 8.3.1 generalizes the Schwarz–Pick Theorem when is chosen to be and is chosen such that .
For the purpose of the proceeding generalization, we define the conformal metric
Its Gaussian curvature is
via the results and definitions in (8.3.1).
Corollary 8.3.1.
Let and suppose is holomorphic, where is a region. For any , define to be a regular metric on with such that
Then ,
for any , where is the metric pullback.
Proof.
Consider the , a conformal metric pullback of to , which satisfies
By Schwarz–Ahlfors–Pick (Theorem 8.3.1), we have
Since , this implies that
Since is a constant,
Corollary 8.3.2 (Generalized Liouville).
If is entire and admits a regular metric of curvature bounded above by a negative constant, then must be constant.
Proof.
By assumption, such that . Then Corollary 8.3.1 gives that
for any . As , . Hence, , implying that . Hence, is constant.
Remark.
Corollary 8.3.2 implies Liouville’s Theorem (Theorem 3.2.3). To justify this differential-geometric generalization, suppose is entire such that is bounded. There then exists some such that . The metric has constant negative curvature on , and hence, under , Corollary 8.3.2 implies that is constant.
It is understood that an entire function is guaranteed to be constant if it is bounded. This is a statement of sufficiency, but it begs the question of the capacity for possible generalization of boundedness under which constancy is still always satisfied.
Consider an entire function , where is an unbounded region such that has positive area. Fix . Then the map maps to a bounded region and hence is constant by Liouville’s Theorem (Theorem 3.2.3), implying the constancy of (the essential proof of Theorem 4.2.4).
In contrast, if is entire and has zero area (one readily considers sets consisting of curves or isolated points), we must be more specific in determining sufficient conditions that still imply constancy of .
Similar to in the proof of the Riemann Mapping Theorem (Theorem 5.3.1), one may use holomorphic square roots or other transformations to reduce to the bounded setting.
Example 8.3.1.
If is entire, then must be constant.
Proof.
Consider the biholomorphism , mapping to . By simple connectivity of , there exists a univalent branch of on . Now omitting the origin, it is trivially realized that . If otherwise, then such that , implying that such that and , implying that and , which does not lie in .
Now fix . By the Open Mapping Theorem (Theorem 5.1.1), such that . Consequently, . Lastly, the function maps to . By Liouville (Theorem 3.2.3), is constant, which implies is constant by the injectivity of , , and .
The preceding examples show that if the omitted set is sufficiently “large” (in the sense of having positive area or disconnecting the plane in certain ways), then any entire function avoiding it must reduce to a constant. However, there are natural limits to the smallness of the omitted set. For instance, the exponential function is an entire non-constant function whose image is , omitting only a single point. Thus, the property that an entire function omits a set is not by itself sufficient to guarantee constancy unless that set is suitably substantial. This observation is formalized by Picard’s Little Theorem (Theorem 8.3.4), which as preluded to before, asserts that any non-constant entire function can omit at most one complex value.
Proposition 8.3.1.
Let be an open set such that contains at least two points. Then admits a conformal metric , such that
for some .
Proof.
Without loss of generality, we may assume that (if not, a linear transformation where are distinct will suffice to transform to such a region).
Define a regular metric with
on .
Since ,
and a similar calculation yields
Hence,
and that
- .
- , .
- .
- .
- in any direction (as in the one-point compactification).
Hence, such that for any and such that for any satisfying . By compactness of and continuity, it attains its supremum of some value by Theorem 1.2.14. Let .
And we have the final implication:
Theorem 8.3.4 (Picard's Little Theorem).
Let be entire such that contains two or more points. Then is constant.
Proof.
By the result of Proposition 8.3.1, we may find a conformal metric on such that satisfying . Then by the aforementioned generalization of Liouville (Corollary 8.3.2), exhibits constancy on and the assertion follows.
Remark.
This is commonly stated in its contrapositive: the image of any non-constant entire function omits at most one value.