Minimization of an area

The problem of finding the minimum of the area of a Riemann surface to which a given domain of the -plane is mapped by a one-to-one regular function of a given class , that is, the problem of finding

(*)

( is the surface element). The integral in (*), taken over , is understood as the limit of integrals over domains , which exhaust the domain , that is, are such that , and such that any closed set lies in from some onwards.

When is the class of functions , , , regular in a given simply-connected domain containing and having more than one boundary point, the minimum of the areas of the images of in the class is given by the unique function univalently mapping onto the full disc , where is the conformal radius of at (cf. Conformal radius of a domain); moreover, .

The problem of finding the minimal area of the image of a multiply-connected domain has also been considered (see [1]).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)