A note on the unique extremality of spiral-stretch maps

The area length method is used to obtain the unique extremal mapping of the affine stretch map, which is the problem of keeping the boundary correspondence from rectangle to quadrangle and satisfying certain initial conditions. As an application, we consider the extreme value problem between two rings, where the outer boundary remains constant and the inner boundary rotates at a certain angle, by using the homotopy method, the Spiral-stretch map is obtained as the unique extremum map.