Insalata di Matematica "The Modulus of a curve family"

Let's try to set up a geometric problem in the Euclidean plane: consider a square Q = (0,1) x (0,1)  and a rectangle R_a,b = (0,a) x (0,b).  The Riemann mapping theorem guarantees that Q and  R_a,b are conformally equivalent. Any conformal homeomorphism between the square and the rectangle extends homeomorphically to the boundary. One might ask whether it is possible to do this in such a way that the horizontal edges of Q are mapped to the corresponding horizontal edges of R_a,b ,and analogously for the vertical edges of the rectangles.

In order to answer such questions we shall introduce the Modulus of a curve family.

This tool has been relevant for the notion of Quasiconformal maps and related theory.