Ahlfors theory

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.

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 connecting these points.

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.

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