Quadratic differential

Local form

Each quadratic differential on a domain U in the complex plane may be written as f(z) ,dz \otimes dz, where z is the complex variable, and f is a complex-valued function on U. Such a "local" quadratic differential is holomorphic if and only if f is holomorphic. Given a chart \mu for a general Riemann surface R and a quadratic differential q on R, the pull-back (\mu^{-1})^*(q) defines a quadratic differential on a domain in the complex plane.

Relation to abelian differentials

If \omega is an abelian differential on a Riemann surface, then \omega \otimes \omega is a quadratic differential.

Singular Euclidean structure

A holomorphic quadratic differential q determines a Riemannian metric |q| on the complement of its zeroes. If q is defined on a domain in the complex plane, and q = f(z) ,dz \otimes dz, then the associated Riemannian metric is |f(z)|(dx^2 + dy^2), where z = x + iy. Since f is holomorphic, the curvature of this metric is zero. Thus, a holomorphic quadratic differential defines a flat metric on the complement of the set of z such that f(z) = 0.

References

• Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp. .
• Y. Imayoshi and M. Taniguchi, M. An introduction to Teichmüller spaces. Translated and revised from the Japanese version by the authors. Springer-Verlag, Tokyo, 1992. xiv + 279 pp. .
• Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp. .