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423:. Martin Schmidt claimed a proof in 2002, but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it). Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as
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Li, Peter; Yau, Shing Tung (1982). "A new conformal invariant and its applications to the
Willmore conjecture and the first eigenvalue of compact surfaces".
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317:) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name
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Langer, Joel; Singer, David (1984). "Curves in the hyperbolic plane and mean curvature of tori in 3-space".
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Marques, Fernando C.; Neves, André (2014). "Min-max theory and the
Willmore conjecture".
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Analele Ştiinţifice ale
Universităţii "Al. I. Cuza" din Iaşi, Secţiunea I a Matematică
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Schmidt, Martin U. (2002). "A proof of the
Willmore conjecture".
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proved the conjecture in the non-embedded case, showing that if
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Lower bound on the integrated squared mean curvature of a torus
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It is not hard to prove that the
Willmore energy satisfies
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The torus with minimal
Willmore energy, with major radius
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Willmore, Thomas J. (1965). "Note on embedded surfaces".
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proved the conjecture in the embedded case, using the
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309:) > 0. In particular, calculation of
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421:Almgren–Pitts min-max theory of minimal surfaces
615:The Bulletin of the London Mathematical Society
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74:was announced in 2012 and published in 2014.
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237:{\displaystyle W(M)=\int _{M}H^{2}\,dA.}
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164:at each point). In this notation, the
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385:{\displaystyle f:\Sigma \to S^{3}}
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665:Theorems in differential geometry
655:Conjectures that have been proved
321:For every smooth immersed torus
427:(by Willmore himself), and for
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538:Math Finds the Best Doughnut
483:10.4007/annals.2014.179.2.6
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435:(by Langer & Singer).
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558:Inventiones Mathematicae
56:. It is named after the
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627:10.1112/blms/16.5.531
460:Annals of Mathematics
413:Fernando Codá Marques
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266:is an embedded round
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68:Fernando Codá Marques
38:differential geometry
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149:principal curvatures
543:The Huffington Post
396:an embedding, then
42:Willmore conjecture
571:10.1007/BF01399507
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347:Peter Wai-Kwong Li
298:for surfaces with
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113:orientable surface
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32:and minor radius 1
121:Riemannian metric
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145:arithmetic mean
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61:mathematician
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64:Tom Willmore
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512:: 493–496.
467:: 683–782.
417:André Neves
123:induced by
72:André Neves
46:lower bound
649:Categories
439:References
433:revolution
115:. Giving
474:1202.6036
425:tube tori
411:In 2012,
370:→
367:Σ
345:In 1982,
274:Statement
206:∫
105:immersion
660:Surfaces
536:(2012) "
635:0751827
579:0674407
518:0202066
491:3152944
147:of the
139:be the
109:compact
58:English
48:on the
26:√
633:
577:
516:
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268:sphere
127:, let
102:smooth
40:, the
594:arXiv
469:arXiv
300:genus
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107:of a
100:be a
54:torus
52:of a
44:is a
429:tori
415:and
349:and
157:and
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70:and
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510:11B
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