Flat metric of a quadratic differential ($d{s}_q=|q{|}^{1/2}$)

For $q=\varphi (z)d{z}^2$, the length element is $|\varphi (z){|}^{1/2}|dz|$, locally Euclidean in flat coordinates. Zeros give cone points, not punctures. Its infimal length in a homotopy class need not determine the hyperbolic length of that class.

The area of the flat metric of a quadratic differential is ${\int }_X|q|$. Multiplying $q$ by a scalar $a$ multiplies this area by $|a|$ and lengths by $|a{|}^{1/2}$. It is preserved by the SL2R action on differentials.