Lecture 15 - Minimal Surfaces
Recall that
Thus we obtain:
For convenience
This is Scherks double periodic surface. For reference (application) check out Randall Kamien (condensed matter physicist).
The Weierstrass Enneper representation
Let
Since
Suppose we differentiate wrt
since
that is:
Similarly we can show that
We know that
That is:
- Catenoid:
- Helicoid:
- Ruled Surface
Let
Suppose
In addition the points
Since we are interested in
Using these we obtain:
Thus if our function is harmonic then it is holomorphic.
Let
Suppose
and that
Note that
Let
will be holomorphic in
Every regular minimal surface has locally an isothermal paramterization of the form equal to: