(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the helicoid and the catenoid are conjugate minimal surfaces

2. Relevant equations

the helicoid is given by the parameterization

X(u,v) = (asinh(v)*cosu, asinh(v)*sinu, au) = (x1, x2, x3)

the catenoid is given by the parameterization

Y(u,v) = (acoshv*cosu, a coshv*sinu, av) = (y1, y2, y3)

3. The attempt at a solution

so i need that d(x1)/du = d(y1)/dv, etc, the component-wise functions must satisfy the Cauchy Riemann equations, but i'm not getting the right answers. clearly for x3 and y3, the C-R equations are satisfied.

i get:

d(x1)/du = -asinhv*sinu

d(y1)/dv = asinhv*cosu

and the C-R equations are not even close to satisfied. but i get that the off terms satisfy the C-R equations,

d(y2)/dv = asinhv*sinu

any help? i know this is a famous classical problem in minimal surfaces, but im so stuck

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# Conjugate minimal surfaces

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