In 1986, M. Elisa G. G. de Oliveira found several new non-orientable minimal surfaces. Here is her projective plane with one planar end and one Enneper end, and finite total curvature –10π:

The Weierstraß representation


G(z)=z3(z2−b2)b2z2−1anddh=−iz(z2−b2)(b2z2−1)(z2−1)4G(z)=z^3\frac{(z^2-b^2)}{b^2z^2-1} \quad \text{and}\quad dh = -\frac{i z \left(z^2-b^2\right) \left(b^2 z^2-1\right)}{\left(z^2-1\right)^4}

contains one parameter that needs to satisfy

b4+10b2−35=0b^4+10 b^2-35=0

in order to solve the single period problem. The two real solutions lead to the surface above, and the two imaginary solutions lead to a surface with more complicated appearance:

Both of these surface comtain the coordinate axes as symmetry lines.

Oliveira has also has a beautiful example with three ends and total curvature -14π:

Resources

  • Mathematica Notebook for 2-ended and for 3-ended cases
  • PoVRay sources for 2-ended and 3-ended cases
  • M. Elisa G. G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proceedings of the American Mathematical Society 1986, Vol. 98, No. 4, pp. 629-636