Conjugate minimal surfaces

  • #1

Homework Statement


Show that the helicoid and the catenoid are conjugate minimal surfaces


Homework 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)


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
 

Answers and Replies

  • #2
here are some others:

1) Prove that no minimal surface can be a compact set
-
Pf.
Let M be a minimal surface that is compact and etry to derive a contradiction.

If M is minimal then the mean curvature is 0 and hence k1 = -k2.

More importantly, any parameterization X of M is continous and hence takes compact sets to compact sets, so our domain of X is a compact set. So X attains its max and min on M. Also every point is a hyperbolic point since the Gaussian curvature is negative.

but im not quite sure where to derive the contradiction.

2) Let F be a mapping from the plane without the origin and the negative real axis. so it's the open half plane onto a surface, we call this half plane U. F is given by a parameterization:

F(u,v) = (u*sinb*cosv, u*sinb*sinv, u*cosb) where b is a constant.
a) Prove that F is a local diffeo of U onto a cone C with the vertex at the origin and 2b has the angle of the vertex.

- I'm not quite sure how to prove that something is a local diffeo. Is this the same as saying that the two spaces are isometric?
 
Last edited:

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