Embedded Delaunay tori and their Willmore energy

TL;DR

This paper constructs a family of embedded rotationally symmetric tori with specific curvature properties, proving their Willmore energy is below a critical threshold, and uses this to solve the isoperimetric-constrained Willmore problem for tori.

Contribution

It introduces a new family of embedded Delaunay tori with controlled Willmore energy, enabling the proof of existence of minimizers under isoperimetric constraints.

Findings

Willmore energy of the tori is strictly below 8π.

Existence of embedded tori minimizing the Willmore functional under isoperimetric constraints.

Construction of spheres with high isoperimetric ratio using Delaunay tori.

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 double round sphere 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 smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay 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.