Can you help me prove that
$2 π^2$ is a lower bound on the Willmore energy of a torus i.e $ $$\iint H^2 \,do≥2 π^2$
differential geometry-Willmore energy
1
$\begingroup$
differential-geometry
-
1https://arxiv.org/abs/1409.7664 – 2017-01-16
-
0What is your definition of "a torus"? If you mean any compact surface of genus $1$, your question is the *Willmore conjecture*; see @WillJagy's link. If you however define a torus as a tube of constant radius $r$ around a circle of radius $R$, you can show that $$H(u,v) = \frac{R+2r \cos u}{2r(R+\cos u)}$$ and hence that $$W(c) = \frac{\pi^2 c^2}{\sqrt{c^2-1}}$$ where $c = R/r$. This function has a unique minimum $2\pi^2 = W(\sqrt{2})$. – 2017-01-18