“Ahlfors theory”的意思、由来-开放百科全书

Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two very first Fields Medals for this theory in 1936.

It can be considered as a generalization of the basic properties of covering maps to the

maps which are "almost coverings" in some well defined sense. It applies to bordered Riemann surfaces equipped with conformal Riemannian metrics.

Preliminaries

A bordered Riemann surface X can be defined as a region on a compact Riemann surface whose boundary ∂X consists of finitely many disjoint Jordan curves. In most applications these curves are piecewise analytic, but there is some explicit minimal regularity condition on these curves which is necessary to make the theory work; it is called the Ahlfors regularity. A conformal Riemannian metric is defined by a length element ds which is expressed in conformal local coordinates z as ds = ρ(z) |dz|, where ρ is a smooth positive function with isolated zeros.

If the zeros are absent, then the metric is called smooth. The length element defines the lengths of rectifiable curves and areas of regions by the formulas

Then the distance between two points is defined as the infimum of the lengths of the curves

Setting and notation

Let X and Y be two bordered Riemann surfaces, and suppose that Y is equipped with a smooth (including the boundary) conformal metric σ(z) dz. Let f be a holomorphic map from X to Y. Then there exists the pull-back metric on X, which is defined by

When X is equipped with this metric, f becomes a local isometry, that is the length of a curve equals to the length of its image. All lengths and areas on X and Y are measured with respect to these two metrics.

If f sends the boundary of X to the boundary of Y, then f is a ramified covering. In particular,

Now suppose that some part of the boundary of X is mapped to the interior of Y. This part is called the relative boundary. Let L be the length of this relative boundary.

First main theorem

The First Main Theorem says that for every regular region and every regular curve,

where L is the length of the relative boundary, and k is the constant that may depend only on

When L = 0 these inequalities become a weak analog of the property a) of coverings.

Second main theorem

Let ρ be the negative of the Euler characteristic (so that ρ = m − 2 for the sphere with m holes). Then

This is meaningful only when ρ(Y) > 0, for example when Y is a sphere with three (or more) holes. In this case, the result can be considered as a generalization of the property b) of coverings.

Applications

Suppose now that Z is an open Riemann surface, for example the complex plane or the unit disc, and let Z be equipped with a conformal metric ds. We say that (Z,ds) is regularly exhaustible if there is an increasing sequence of bordered surfaces D_j contained in Z with their closures, whose union in Z, and such that

Ahlfors proved that the complex plane with arbitrary conformal metric is regularly exhaustible. This fact, together with the two main theorems implies Picard's theorem, and the

Second main theorem of Nevanlinna theory. Many other important generalizations of Picard's

One especially striking result (conjectured earlier by André Bloch) is the Five Island theorem.

Five-island theorem

Let D₁,...,D₅ be five Jordan regions on the Riemann sphere with disjoint closures. Then there exists a constant c, depending only on these regions, and having the following property:

Let f be a meromorphic function in the unit disc such that the spherical derivative satisfies

Then there is a simply connected region G contained with its closure in the unit disc, such

This does not hold with four regions. Take, for example f(z) = ℘(Kz), where K > 0 is arbitrarily large, and ℘ is the Weierstrass elliptic function satisfying the differential equation

All preimages of the four points e₁,e₂,e₃,∞ are multiple, so if we take four discs with disjoint closures around these points, there will be no region which is mapped on any of these discs homeomorphically.

Remarks

Simple proof of the Five Island Theorem, not using the Ahlfors theory, was developed by Bergweiler.^[7]

References

1. ^{{cite paper|first=L.|last= Ahlfors|authorlink=Lars Ahlfors|title=Zur Theorie der Uberlagerungsflachen|journal=Acta Mathematica|year=1935|volume=65|pages=157–194 (German)}}
2. ^{{cite book | first=W.|last= Hayman|authorlink=Walter Hayman|title=Meromorphic functions|publisher=Oxford University Press|year=1964}}
3. ^{{cite book| first=R.|last= Nevanlinna|authorlink=Rolf Nevanlinna|title=Analytic functions|publisher=Springer Verlag|year=1970}}
4. ^{{cite book|first=M.|last=Tsuji| title=Potential theory in modern function theory|publisher=Maruzen|place=Tokyo|year=1959}}
5. ^{{cite paper|first=Yukinari|last=Toki|title=Proof of Ahlfors principal covering theorem|journal=Rev. Math. Pures Appl.|volume= 2|year= 1957 |pages=277–280.}}
6. ^{{cite paper|first=Henry|last=de Thelin|title=Une démonstration du théorème de recouvrement de surfaces d'Ahlfors|journal=Ann. Fac. Sci. Toulouse Math.|year=2005|volume=51|pages=203–209. (French)}}
7. ^{{cite paper|first=W.|last= Bergweiler|title=A new proof of the Ahlfors five islands theorem|journal=J. Anal. Math.|volume=76|year=1998|pages=337–347.}}