Quadratic differential

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

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

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

• Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp. ISBN 3-540-13035-7.
• 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. ISBN 4-431-70088-9.
• Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp. ISBN 0-471-84539-6.