On the Existence of Certain General Extremal Metrics

Noncompleteness of the Weil-Petersson metric for Teichmüller space

“…The existence of such differentials on finite Riemann surfaces is a consequence of the solution by J. Jenkins of a class of free homotopy a conmodule problems, [7]. A. Douady and J. Hubbard recently confirmed jecture of K. Strebel that such differentials represent a dense subset of the space of all analytic quadratic differentials, [6].…”

On metrics defined by modules

“…The module of this homotopy class is equal to the maximal module of a domain such as D{x, y). This follows from the present author's fundamental theorem [5]. There is a homotopy class %* (unique except in certain special cases) for which the module is maximal.…”

Guiding isotopies and holomorphic motions

“…The existence and uniqueness of q with these properties is well-known, see [30,19,15,12]. For any simple closed curve α not homotopically trivial and not homotopic to a puncture on any hyperbolic Riemann surface, such a holomorphic quadratic differential is obtained by maximizing the modulus of a cylinder among all cylinders on the surface homotopic to α.…”

“…In the generic case, when the critical graph has only three pronged singularities, there are 2(n + 1) intervals that partition the circle {w : |w| = 1}. These intervals are pairwise identified so as to realize the critical graph with n + 1 edges, where n is the cardinality of E. For a proof of the existence of q with these properties see [24] and [19]. We put q equal to the negative of this quadratic differential.…”