A conjugate minimal surface is a minimal surface whose coordinate functions are harmonic conjugates of those of a given conformally parametrized minimal surface on a simply connected domain (Meeks and Pérez 2012, p. 21).
More explicitly, let 
be an analytic map
into 
with 
and ![[F_1^'(z)]^2+[F_2^'(z)]^2+[F_3^'(z)]^2=0](/images/equations/ConjugateMinimalSurface/Inline4.svg)
. The immersion
 |
has conjugate immersion
 |
The two immersions induce the same first fundamental form, so they are locally isometric. The conjugate is determined up to a translation once the sign convention for harmonic conjugates is fixed.
On a domain that is not simply connected, the imaginary periods of the defining integral may fail to vanish even when its real periods vanish. A single-valued conjugate can instead be defined on the universal cover. The catenoid, considered on its universal cover, and the helicoid form a standard conjugate pair (Meeks and Pérez 2012, pp. 21 and 27).
