Extremal length lower bound from a closed one-form ($\lambda (\gamma ,X)\ge ({\int }_{\gamma }\alpha {)}^2/{\int }_X|\alpha {|}^{2}dA$)

Let $\alpha$ be a real closed differential form on a Riemann surface, of finite positive conformal energy $E=\int |\alpha {|}^{2}dA$. If its period on a loop homotopy class $\gamma$ is $p$, then $\lambda (\gamma ,X)\ge p^2/E$. Indeed the conformal metric $\rho =|\alpha |$ has area $E$ and each loop in the class has length at least $|\int \alpha |=|p|$. Energy is independent of the auxiliary smooth conformal metric used to compute the pointwise norm: the inverse scaling of the squared norm cancels the area scaling.