On the existence and uniqueness of Strebel differentials

Abstract

Let X be a compact Riemann surface of genus g > 1, and q GH°(X> £2® ) be a holomorphic quadratic form on X. A tangent vector £ G Tx X is called horizontal if (q, £ ® £> > 0. The horizontal vectors define a foliation of X singular at the zeroes of q. The form q is called a Strebel form if the leaves of this foliation are compact. If q is a Strebel form, the leaves of the foliation through a zero of q form a graph I , and X Ta is a union of metric straight cylinders (for the metric |#| ). The central circles in each cylinder form a set of disjoint, nonpairwise homotopic and homotopically nontrivial simple closed curves on X, called the system of curves associated to q. Let M be an oriented differentiable compact surface of genus g, and C a system of n simple closed curves on M, disjoint, not pairwise homotopic and homotopically nontrivial. In the vector bundle Q of pairs (0, q), with 0 in the Teichmiiller space &M (see [2] for notation) and q a quadratic form on the Riemann surface above 0, consider the space Ec C Q of Strebel forms whose associated system of curves is homotopic to C. Denote n: Ec —• 0 M x R+ the map whose first factor is the canonical projection, and whose second factor gives the heights of the cylinders. Our main result is the following