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

  • Andrea Mondino ORCID logo EMAIL logo and Christian Scharrer ORCID logo


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 8 π . 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.

MSC 2010: 53A05; 53A30

Communicated by Tristan Rivière

Award Identifier / Grant number: 802689

Award Identifier / Grant number: EP/HO23364/1

Funding statement: The first author is supported by the European Research Council (ERC), under the European’s Union Horizon 2020 research and innovation programme, via the ERC Starting Grant “CURVATURE”, grant agreement no. 802689. The second author is supported by the EPSRC as part of the MASDOC DTC at the University of Warwick, grant no. EP/HO23364/1.


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Received: 2021-01-04
Revised: 2021-04-30
Accepted: 2021-05-05
Published Online: 2021-06-01
Published in Print: 2023-07-01

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