Localized deformation of Riemann surfaces
Let \(Y\) be a compact bordered Riemann surface inside an analytically finite Riemann surface \(X\). Each complex deformation of \(Y\) induces naturally a complex deformation of \(X\). In this way, one has a natural holomorphic map \(\phi\) from the deformation space of \(Y\), say \(D(Y)\), to the Teichmüller space of \(X\). In this article, the author proves that, if \(X\) is analytically finite hyperbolic Riemann surface, then: (i) the image \(\phi(D(Y))\) is an open set in the Teichmüller space of \(X\); (ii) \(\phi(D(Y))\) is relatively compact if and only if \(Y\) can be homotoped to either a point or to a punctured disc in \(X\); (iii) \(\phi\) is surjective if and only if \(X-Y\) is disjoint union of discs and/or punctured discs. In particular, the author observes that the deformations of an analytically finite hyperbolic Riemann surface induced by the Schiffer variations remain inside a compact part of Teichmüller space of the surface.
scientific article; zbMATH DE number 2186478