I was just wondering whether anyone could tell me more about the Enneper surface and its history (why it is important historically in the development of mathematics), or where to go in order to learn more.
Also, while reading Wikipedia, I saw the term "Enneper-like surfaces" (for higher dimensions). I was wondering what "Enneper-like" meant. How "like" is it? Is there a unique generalization of the Enneper surface for each dimension, and in what sense are these generalizations? What theorems have been proven for these "like" surfaces, and why do we only know of specific examples up to seven dimensions?
Any more cool information that you would like to share would be appreciated too!
$$ X(u,v) = \Re\left(\int { 1 - z^2} dz \right)$$
$$\Re\left( { z - \frac{1}{3}z^3} \right)$$
$$\Re\left( { (u + i v) - \frac{1}{3}(u + i v)^3 } \right)$$
$$ u - \frac{1}{3} u^3 + uv^2 $$
$$ Y(u,v) = \Re\left(\int i({ 1 + z^2}) dz \right)$$
$$\Re\left( i ({ z + \frac{1}{3}z^3}) \right)$$
$$\Re\left( i ({ z + \frac{1}{3}z^3}) \right)$$
$$\Re\left( { i(u + i v) + \frac{1}{3}i(u + i v)^3 } \right)$$
$$ -v -u^2 v + \frac{1}{3} v^3 $$
$$ Z(u,v) = \Re\left(\int { 2z } dz \right)$$
$$\Re\left( { z^2 } \right)$$
$$ u^2 - v^2 $$
So we can represent the surface using the patch $ (u− \frac{1}{3} u^3 + uv^2,−v −u^2v + \frac{1}{3} v^3 ,u^2 − v^2) $ with $ u, v \in R $
See: en.wikipedia.org/wiki/Weierstrass–Enneper_parameterization