Total mean curvature of an immersed torus.

Question

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ over the immersed manifold, where H is the mean curvature. I don't have any idea about it.

Answer 1

For every smooth immersed torus $M$ in $\mathbb{R}^3$

$$\int_MH^2dA≥ 2π^2$$.

http://en.wikipedia.org/wiki/Willmore_conjecture