Some problems in conformal mapping

Abstract

Thus p = l does give an extremal metric as stated and any other extremal metric would be almost everywhere equal to 1.We observe that, in this case, the greatest lower bound is actually a minimum.The extremal length of the above class of curves we may call the extremal distance of the sides 12 and 34.As we have seen, it is i/l where / refers to the corresponding rectangle.That a conformally invariant meaning can be attached to I shows, as has long been known, that the ratio of the sides of a rectangle on which a given quadrangle can be mapped is uniquely determined.4. Now let us suppose we have a second quadrangle Q' with vertices 1', 2', 3', 4' taken in counterclockwise order on the contour as before.Let it be mapped on a rectangle in the z'-plane having sides /' and 1 in place of / and 1.We ask under what conditions the second quadrangle can be mapped conformally into the first in such a way that the side 1'2' lies along the side 12