The heights theorem for quadraic differentials on Riemann surfaces
The authors study quadratic differentials \(\phi\) dz\({}^ 2\) on a Riemann surface \({\mathcal R}\), the former being restricted to have only finite critical points (zeros and simple poles, with the surface assumed punctured at the latter for homotopy purposes). Their principal result is what they call the heights theorem. By the height of a Jordan contour on \({\mathcal R}\) is meant the greatest lower bound of \(\int | \vartheta \sqrt{\phi}dz|\) taken on (freely) homotopic Jordan contours. The theorem says that if two quadratic differentials have the same height for every class of Jordan contours they coincide provided that the surface is parabolic and the differentials have finite norm. It can be extended in an appropriate manner to bordered surfaces. They give applications to the problem of approximation of quadratic differentials by those corresponding to extremal imbedding of a doubly-connected domain in the surface. The existence of such an approximation is well known for compact surfaces. In more general cases they obtain such results only by imposing subsidiary conditions. Finally they indicate some relationships to extremal quasiconformal mappings.
approximation of quadratic differentials
scientific article; zbMATH DE number 3923081