Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the value of $$\frac{\int_\mathcal{M} K^2dA\cdot A(\mathcal{M})}{\left(\int_\mathcal{M}KdA\right)^2}$$ for the Gaussian curvature $K$. Ideally, I want to find a lower bound on this term. By Cauchy-Schwarz, we know that this term must be bigger than 1. Gauß-Bonnet then lets us deduce that $\int_\mathcal{M} KdA=2\pi\chi(\mathcal{M})=2\pi(2-2g)=-8\pi$. Therefore, my question boils down to asking, whether we can bound the squared Gaussian curvature integral $\int_\mathcal{M}K^2dA$ and the surface area $A(\mathcal{M})$ from below, or whether we can express these formulae in terms of each other? That would be incredibly helpful!
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