Embedded Delaunay tori and their Willmore energy

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-Rivière 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.

arXiv comment28 pages. Final version to appear in Nonlinear Analysis (TMA)

References in corpus (7)

• Willmore minimizers with prescribed isoperimetric ratio
• Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint
• Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy
• The Willmore Flow of Tori of Revolution
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• Delaunay Surfaces
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Cited by in corpus (2)

• Li-Yau inequalities for the Helfrich functional and applications
• The Willmore flow with prescribed isoperimetric ratio