View source for Weierstrass criterion

''for a minimal surface''

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" />.



====Comments====


====References====
<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>