Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.
When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and $\int_{\Sigma}\text{H}^2dV\geq4\pi$.
When $g=1$, we have $\int_{\mathbb T^2}\text{H}^2dV\geq2\pi^{2}$ and $\int_{\Sigma}\text{H}^2dV\geq2\pi^{2}$.
I want to ask that these theorems can be generalized to a compact surface with $g>1$ in $\mathbb E^3$? And the statement below is right or not?
$$\text{When}~~g(\Sigma_{1})>g(\Sigma_{2})\text{, then do we have} \int_{\Sigma_{1}}\text{H}^2dV\geq\int_{\Sigma_{2}}\text{H}^2dV\text{?}$$
