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

TL;DR

This paper investigates the minimal Willmore functional values for tori embedded in specific product Riemannian manifolds, revealing exact minima in some cases and non-existence in others, advancing understanding of geometric optimization in these spaces.

Contribution

It provides explicit minimal Willmore functional values for tori in certain product manifolds, including new results for $S^2\times S^1$, $R^2\times S^1$, and $H^2(-c)\times S^1$, and discusses conditions for minimality.

Findings

Minimum of W(T^2) is 0 in S^2×S^1.

No torus minimizes W in R^2×S^1.

Minimum of W(x) is 2π^2√c in H^2(-c)×S^1.

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 $2\pi^2\sqrt{c}$.