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| In [[differential geometry]] in [[mathematics]] the '''Willmore conjecture''' is a [[conjecture]] about the [[Willmore energy]] of a [[torus]], named after the [[England|English]] [[mathematician]] [[Tom Willmore]].<ref>{{cite journal|last=Willmore|first=Thomas J.|title=Note on embedded surfaces|journal=An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.)|year=1965|volume=11B|pages=493-496|url=http://www.ams.org/mathscinet-getitem?mr=0202066}}</ref>
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| ==Willmore energy==
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| Let ''v'' : ''M'' → '''R'''<sup>3</sup> be a [[smooth function|smooth]] [[immersion (mathematics)|immersion]] of a [[compact space|compact]], [[orientability|orientable surface]]. Giving ''M'' the [[Riemannian metric]] induced by ''v'', let ''H'' : ''M'' → '''R''' be the [[mean curvature]] (the [[arithmetic mean]] of the [[principal curvature]]s ''κ''<sub>1</sub> and ''κ''<sub>2</sub> at each point). In this notation, the '''Willmore energy''' ''W''(''M'') of ''M'' is given by
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| :<math> W(M) = \int_M H^2 \, dA. </math>
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| It is not hard to prove that the Willmore energy satisfies ''W''(''M'') ≥ 4''π'', with equality [[if and only if]] ''M'' is an embedded round [[sphere]].
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| ==The conjecture==
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| Calculation of ''W''(''M'') for a few examples suggests that there should be a better bound than ''W''(''M'') ≥ 4''π'' for surfaces with [[genus (topology)|genus]] ''g''(''M'') > 0. In particular, calculation of ''W''(''M'') for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name
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| : For every smooth immersed torus ''M'' in '''R'''<sup>3</sup>, ''W''(''M'') ≥ 2''π''<sup>2</sup>. | |
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| In 2012, a proof of the conjecture in a paper of [[Fernando Codá Marques]] and [[André Neves]] appeared using the min-max theory of [[minimal surface]]s.<ref>{{cite journal |last=Marques |first=Fernando C. |first2=André |last2=Neves |title=Min-max theory and the Willmore conjecture |journal=Annals of Mathematics |year=2013 |id={{arXiv|1202.6036}} }}</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Conjectures]]
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| [[Category:Differential geometry]]
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| [[Category:Surfaces]]
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| [[de:Willmore-Energie]]
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Revision as of 16:06, 13 February 2014
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