Quadratic Differentials of General Type

Abstract

For any rectifiable closed curve γ on a Riemann surface R and holomorphic quadratic differential φ on R the φ-length

$$ {\left| \gamma \right|_\varphi } = {\int\limits_\gamma {\left| {\varphi (z)} \right|} ^{\frac{1}{2}}}\left| {dz} \right| $$

is defined. Rectifiability is meant with respect to the local parameters z on R; it is evidently independent of the choice of the local parameter. The infimum of the φ-lengths for all loops $\tilde \gamma$ in the free homotopy class of γ is denoted by

$$ {l_\varphi }(\gamma ) = \mathop {\inf }\limits_{\tilde \gamma \sim \gamma } \int\limits_\gamma {{{\left| {\varphi (z)} \right|}^{\frac{1}{2}}}} \left| {dz} \right| $$

.