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The Willmore Conjecture

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Abstract

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory.

In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field.

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Acknowledgements

The first author is grateful to École Polytechnique, École Normale Supérieure and Université Paris-Est (Marne-la-Vallée) for the hospitality during the writing of this paper.

Both authors are very grateful to Robert Bryant and Robert Kusner for their various remarks and suggestions that improved the exposition of the paper.

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Authors and Affiliations

  1. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Fernando C. Marques

  2. Imperial College, Huxley Building, 180 Queen’s Gate, London, SW7 2RH, United Kingdom

    André Neves

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  1. Fernando C. Marques
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  2. André Neves
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Correspondence to André Neves.

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The first author was partly supported by CNPq-Brazil and FAPERJ. The second author was partly supported by Marie Curie IRG Grant and ERC Start Grant.

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Marques, F.C., Neves, A. The Willmore Conjecture. Jahresber. Dtsch. Math. Ver. 116, 201–222 (2014). https://doi.org/10.1365/s13291-014-0104-8

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