On the Willmore functional of 2-tori in some product Riemannian manifolds

Abstract

We discuss the minimum of Willmore functional of torus in a Riemannian manifold $N$, especially for the case that $N$ is a product manifold. We show that when $N=S^2\times S^1$, the minimum of $W(T^2)$ is 0, and when $N=R^2\times S^1$, there exists no torus having least Willmore functional. When $N=H^2(-c)\times S^1$, and $x=\gamma\times S^1$, the minimum of $W(x)$ is $2\pi^2\sqrt{c}$.