Embedded Delaunay tori and their Willmore energy. (October 2022)

• Record Type: Journal Article
• Authors: Scharrer, Christian
• Is Part Of: Nonlinear analysis. Volume 223(2022)
• Publication Date: 2022-10
• Subjects: Willmore functional -- Helfrich functional -- Delaunay surfaces -- Isoperimetric ratio -- Tori of revolution -- Constant mean curvature surfaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113010
• Languages: English
• ISSNs: 0362-546X

Abstract

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 π . 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.