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

TL;DR

This paper establishes a strict inequality relating the Willmore energies of surfaces and their connected sum under isoperimetric constraints, leading to existence results for minimizers in the constrained Willmore problem across all genus levels.

Contribution

It proves a new strict inequality for the Willmore functional under isoperimetric constraints, extending previous work and ensuring minimizer existence below a specific energy threshold.

Findings

Strict inequality between Willmore energies of surfaces and their connected sum.

Existence of minimizers for the isoperimetric constrained Willmore problem in all genus.

Minimizers exist when minimal energy is strictly below 8π.

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-Rivi\`ere, 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.