*preprint*

**Inserted:** 10 feb 2021

**Year:** 2021

**Abstract:**

A family of embedded rotationally symmetric tori in the Euclidean $3$-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a round sphere of multiplicity $2$ is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8\pi$. Combining such a strict inequality with previous works by Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smooth embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Moreover, because of their symmetry, the tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.