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en>Theemathas: RMQ link

2014-11-30T06:36:02Z

RMQ link

en>Gareth Jones: /* References */ add doi link

2014-02-13T14:06:21Z

References: add doi link

← Older revision		Revision as of 16:06, 13 February 2014
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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>~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is exactly where her home is but her spouse wants them to move. What me and my family members adore is to climb but I'm considering on starting something new. Credit authorising is how she makes a residing.<br><br>my web blog ... [http://www.chk.woobi.co.kr/xe/?document_srl=346069 free online tarot card readings]

	~~==Willmore energy==~~

	~~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~~

	~~:<math> W(M) = \int_M H^2 \, dA~~. ~~</math>~~

	~~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]]~~.

	~~==The conjecture==~~

	~~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

	: ~~For every smooth immersed torus ''M'' in '''R'''<sup>3<~~/~~sup>, ''W''(''M'') ≥ 2''π''<sup>2<~~/~~sup>~~.

	~~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>~~

	==~~References==~~
	~~{{reflist}}~~

	~~[[Category:Conjectures]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Surfaces]]~~

	~~[[de:Willmore-Energie]~~]

223.180.250.35: /* Generalization for higher dimensions */

2014-01-31T05:11:56Z

Generalization for higher dimensions

← Older revision		Revision as of 07:11, 31 January 2014
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	~~Wilber Berryhill~~ is the title ~~his parents gave him and he totally digs that name~~. ~~Distributing production is where her primary earnings comes from~~. ~~What me and my family adore is performing ballet but~~ I~~'ve been using on new things recently~~. ~~My wife and I reside in Mississippi but now~~ I'~~m considering other choices.~~<br><br>~~Take~~ a ~~look at my web page best psychic readings~~ ([~~http~~://~~jplusfn~~.~~Gaplus~~.kr/xe/~~qna~~/~~78647 jplusfn~~.~~Gaplus~~.kr])		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>

			==Willmore energy==

			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

			:<math> W(M) = \int_M H^2 \, dA. </math>

			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]].

			==The conjecture==

			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

			: For every smooth immersed torus ''M'' in '''R'''<sup>3</sup>, ''W''(''M'') ≥ 2''π''<sup>2</sup>.

			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>

			==References==
			{{reflist}}

			[[Category:Conjectures]]
			[[Category:Differential geometry]]
			[[Category:Surfaces]]

			[[de:Willmore-Energie]]

en>Dicklyon: Reverted edits by Phishman3579 (talk) to last version by 217.124.190.4

2012-06-13T21:17:39Z

Reverted edits by Phishman3579 (talk) to last version by 217.124.190.4

New page

Wilber Berryhill is the title his parents gave him and he totally digs that name. Distributing production is where her primary earnings comes from. What me and my family adore is performing ballet but I've been using on new things recently. My wife and I reside in Mississippi but now I'm considering other choices.<br><br>Take a look at my web page best psychic readings ([http://jplusfn.Gaplus.kr/xe/qna/78647 jplusfn.Gaplus.kr])