= Extremal length lower bound from a closed one-form
{title2=$\lambda(\gamma,X)\geq(\int_\gamma\alpha)^2/\int_X|\alpha|^2\,dA$}
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 $\lambda(\gamma,X)\geq p^2/E$. 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.
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