Prime ends

The prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding a circle in an appropriate way.

The concept of prime end was introduced by Carathéodory to describe the boundary behavior of conformal maps in the complex plane in geometric terms. The theory has been generalized to more general open sets, too.

Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can expressed as follows:

If f maps the unit conformally and one-to-one onto the domain B, it induces a one-to-one mapping between the points on the unit circle and the prime ends of B.

The set of prime ends of the domain B is the set of equivalent classes of chains of arcs converging to a point on the boundary of B. In this way, a point in the boundary may correspond to many points in the prime ends of B, and conversely, many points in the boundary may correspond to a point in the prime ends of B. (for more precise definition of "chains of arc" and their equivalent classes see the references).

The expository paper of Epstein provides a good account of this theory with complete proofs. It also introduces a definition which make sense any open set in and dimension.

References

C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913), 323–370.

D. B. A. Epstein, Prime ends, Proc. london math soc. 3 42, 1981, 385-414.

John N Mather, Topological proof of some purely topological consequences of Caratheodory’s theory of prime ends, Selected Studies, Th. M. Rassias, G. M. Rassias, eds. Amsterdam: North Holland, 1982, 225-255.

Christian Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag 1992.

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