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Difference between revisions of "Weierstrass criterion"

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''for a minimal surface''
 
''for a minimal surface''
  
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For a two-dimensional surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974201.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974203.png" />, with [[Isothermal coordinates|isothermal coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974205.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974206.png" />, to be minimal (cf. [[Minimal surface|Minimal surface]]), it is necessary and sufficient that the components of its position vector be harmonic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097420/w0974207.png" />.
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For a two-dimensional surface in $n$-dimensional Euclidean space $E^n$, $n\geq3$, with [[Isothermal coordinates|isothermal coordinates]] $u$ and $v$ of class $C^2$, to be minimal (cf. [[Minimal surface|Minimal surface]]), it is necessary and sufficient that the components of its position vector be harmonic functions of $(u,v)$.
  
  
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====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C.C. Nitsche,  "Vorlesungen über Minimalflächen" , Springer  (1975)  pp. §455</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Weierstrass,  "Math. Werke" , '''3''' , G. Olms, reprint  (1967)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C.C. Nitsche,  "Vorlesungen über Minimalflächen" , Springer  (1975)  pp. §455 {{ZBL|0319.53003}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Weierstrass,  "Math. Werke" , '''3''' G. Olms [1903] {{ZBL|34.0023.01}} reprint  (1967) </TD></TR>
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Latest revision as of 18:42, 30 October 2017

for a minimal surface

For a two-dimensional surface in $n$-dimensional Euclidean space $E^n$, $n\geq3$, with isothermal coordinates $u$ and $v$ of class $C^2$, to be minimal (cf. Minimal surface), it is necessary and sufficient that the components of its position vector be harmonic functions of $(u,v)$.


Comments

References

[a1] J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455 Zbl 0319.53003
[a2] K. Weierstrass, "Math. Werke" , 3 G. Olms [1903] Zbl 34.0023.01 reprint (1967)
How to Cite This Entry:
Weierstrass criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_criterion&oldid=14513
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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