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Weierstrass criterion

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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=42230
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article