A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint

Abstract

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivière, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8\pi$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.

arXiv comment16 pages. Final version to appear in Advances in Calculus of Variations

References in corpus (4)

• 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
• Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

Cited by in corpus (3)

• Li-Yau inequalities for the Helfrich functional and applications
• Embedded Delaunay tori and their Willmore energy
• The Willmore flow with prescribed isoperimetric ratio