Weierstrass criterion

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 $(u,v)$.

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)