en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB

2013-03-08T06:38:48Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB

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[[File:Chen-Gackstatter-Thayer surfaces.png|thumb|The first nine Chen–Gackstatter surfaces.]]

In [[differential geometry]], the '''Chen–Gackstatter surface family''' (or the '''Chen–Gackstatter–Thayer''' surface family) is a family of [[minimal surfaces]] that generalize the [[Enneper surface]] by adding handles, giving it nonzero [[Genus (mathematics)|topological genus]].<ref>{{citation|last1=Chen|first1=C. C.|last2=Gackstatter|first2=F.|title=Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ|journal=Math. Ann.|volume=259|pages=359–369|year=1982|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002321963&IDDOC=129137}}</ref><ref name="thayer">{{citation|first=Edward C.|last=Thayer|title=Higher-genus Chen–Gackstatter surfaces and the Weierstrass representation for surfaces of infinite genus|journal=Experiment. Math.|volume=4|issue=1|year=1995|pages=19–39|url=http://projecteuclid.org/euclid.em/1062621140}}</ref>

They are not [[Topological embedding|embedded]], and have Enneper-like [[End (topology)|ends]]. The members <math>M_{ij}</math> of the family are indexed by the number of extra handles ''i'' and the winding number of the Enneper end; the total genus is ''ij'' and the total [[Gaussian curvature]] is <math>-4\pi(i+1)j</math>.<ref>{{mathworld|author=Barile, Margherita|title=Chen–Gackstatter Surfaces|urlname=Chen-GackstatterSurfaces}}</ref> It has been shown that <math>M_{11}</math> is the only genus one orientable complete minimal surface of total curvature <math>-8\pi</math>.<ref>{{citation|last=López|first=F. J.|title=The classification of complete minimal surfaces with total curvature greater than −12''π''|journal=Trans. Amer. Math. Soc.|volume=334|pages=49–73|year=1992}}.</ref>

It has been conjectured that continuing to add handles to the surfaces will in the limit converge to the [[Scherk_surface#Scherk.27s_second_surface|Scherk's second surface]] (for ''j'' = 1) or the [[saddle tower]] family for ''j'' > 1.<ref name="thayer" />

==References==

{{reflist}}

==External links==
* The Chen–Gackstatter Thayer Surfaces at the Scientific Graphics Project [https://secure.msri.org/about/sgp/jim/geom/minimal/library/chengack/index.html]
* Chen–Gackstatter Surface in the Minimal Surface Archive [http://www.indiana.edu/~minimal/archive/Tori/Tori/Chen-Gackstatter/web/index.html]
* Xah Lee's page on Chen–Gackstatter [http://xahlee.info/surface/chen_gackstatter/chen_gackstatter.html]

{{DEFAULTSORT:Chen-Gackstatter surface}}
[[Category:Minimal surfaces]]

Henneberg surface - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB