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Homework Help: Conjugate minimal surfaces

  1. Mar 26, 2007 #1
    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
     
  2. jcsd
  3. Mar 26, 2007 #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: Mar 26, 2007
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