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Let $X$ and $Y$ be isothermal parametrizations of minimal surfaces such that their component functions are pairwise harmonic conjugates, then $X$ and $Y$ are called conjugate minimal surfaces.

My question is: Are the helicoid and the catenoid conjugate minimal surfaces? It seems to be impossible after a short calculation.

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1 Answer 1

Yes, they are conjugate minimal surfaces.

I remember running into calculation errors when I first did this problem, too. The trick that worked for me was to rotate the helicoid by an angle of $\frac{\pi}{2}$. Hopefully you should still have isothermal coordinates (check this), but now the Cauchy-Riemann Equations will be satisfied (check this too).

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One way to look at this $\pi / 2$ rotation is as a complex rotation in the Weierstrass parametrization. – Willie Wong Apr 6 '11 at 16:17
That's pretty cool. – Jesse Madnick Apr 6 '11 at 16:21

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