User:Anders Sandberg/sandbox

http://repositories.tdl.org/ttu-ir/bitstream/handle/2346/9226/31295017082289.pdf?sequence=1

A parametrization $$r(u,v): \R^2 \rightarrow \R^3$$ of a surface is said to be isothermal if $$r_u \cdot r_u = r_v \cdot r_v \neq 0$$ and $$ r_u \cdot r_v = 0$$. This maps the $$u,v$$ plane onto the surface in a conformal way. There always exist local isothermal parametrizations of a minimal surface. For any surface, $$\nabla^2 r = 2EHN$$ where $$E=r_u \cdot r_u$$, $$H$$ is the mean curvature and $$N$$ is the normal vector of the surface. Hence if $$\nabla^2 r =0$$ $$H$$ must vanish since $$E$$ and $$N$$ are nonzero: a harmonic isothermal parametrization implies zero mean curvature. A surface is minimal iff such a parametrization exists.

Instead of seeing the map as a map from $$\R^2$$ to $$\R^3$$ it can be seen as a map $$r(z): \Complex \rightarrow \R^3$$, where $$z=u+vi$$ and $$r(z)=(r^1(z),r^2(z),r^3(z))$$. Let $$\varphi^j = \partial r^j/dz$$ be the complex derivative of the components of $$r$$. $$\varphi_j$$ is analytic iff $$r^j$$ is harmonic, and the combined map of such analytic functions implies a harmonic isothermal parametrization of the surface.

http://books.google.co.uk/books?id=9YhBOg6vO-EC&pg=PA155&dq=bour's+minimal+surface+weierstrass&hl=en&sa=X&ei=FEaNUMyBCenG0QXizoDIAQ&ved=0CC8Q6AEwAA#v=onepage&q=bour's%20minimal%20surface%20weierstrass&f=false

http://www3.mathematik.tu-darmstadt.de/fileadmin/home/users/12/mfl.pdf

Alternative versions

A common alternative form is:
 * $$\begin{align}

\varphi_1 &{}= f(1/g-g)/2 \\ \varphi_2 &{}= \mathbf{i} f(1/g+g)/2 \\ \varphi_3 &{}= f \end{align}$$ where f is holomorphic and g meromorphic, and at each zero or pole of order k of g the function f has a zero of order at least k.

This makes the link to the geometrical properties more clear, see

Another form describes the surface using just one holomorphic function f:
 * $$\begin{align}

\varphi_1 &{}= (1-z^2)f \\ \varphi_2 &{}= \mathbf{i} (1+z^2)f \\ \varphi_3 &{}= 2zf \end{align}$$

Some parametrisations (note that there may exist several alternate parametrizations of the same surface):

Using the lattice $$\{1/2, i/2\}$$:
 * $$\begin{align}

X(z) &= \Re \left \{ \frac{1}{2} \left (-\zeta(z) + \pi(z-i) +\frac{\pi^2(1+i)}{4e_1} \right ) + \frac{\pi}{4e_1}\left (\zeta(z-\frac{1}{2}) - \zeta(z-\frac{i}{2}) \right)   \right \}\\ Y(z) &= \Re \left  \{ \frac{i}{2} \left ( -\zeta(z) - \pi(z-1) - \frac{\pi^2(1+i)}{4e_1} \right ) - \frac{i\pi}{4e_1}\left ( \zeta(z-\frac{1}{2}) - \zeta(z-\frac{i}{2}) \right ) \right \}\\ Z(z) &= \frac{\sqrt{2\pi}}{4} \Re \left \{ \ln \left ( \frac{\wp(z) - e_1}{\wp(z)+e_1} \right ) - i\pi \right \}\\ \end{align}$$

where $$e_1 = \wp(1/2;g_2,g_3)\approx 6.87519$$, where $$g_2, g_3$$ are the invariants for the lattice.

A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., Boca Raton, FL: CRC Press, 1999.

http://www.mathematica-journal.com/data/uploads/2010/12/Melko.pdf http://stanwagon.com/wagon/Misc/HTMLLinks/InvisibleHandshake_5.html

http://tpfto.wordpress.com/2012/05/14/a-short-note-on-costas-minimal-surface/


 * (Survey of current minimal surface theory, with historical notes and discussion of the definition of minimal surfaces.)


 * (Review of minimal surface theory, in particularly boundary value problems. Contains extensive references to the literature.)