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

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 and Rivière, our strict inequality leads to existence of minimizers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below 8π. Besides the geometric interest, such a minimization problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.