Application of generalized Implicit Hypersurface theory indicates that the Hopf
map generates a 2D surface in 4D that has two principal curvatures which are
pure imaginary conjugates, and one zero curvature (which reduces the 3D implicit
surace to a 2D surface.)
This means that the the implicit Hopf surface is a Minimal Surface with Positive
Gauss Curvature. The usual minimal surface has negative (or zero) Gauss
curvature. Minimal surfaces are usually associated with soap films of negative
Gauss curvature. Soap films are know to have zero pressure gradients normal to
the surface, and support tangential discontinuities such as are seen in
hydrodynamic wakes. The negative Gauss curvature surfaces are not compact
without boundary.
However, with imaginary principal curvatures, it is possible to produce a
compact surface with minimal surface properties! The applications to compact
biological membranes is immediate. They are implicit surface projections from
4D to 2D and would not exist as implicit surfaces in 3D.
The Falaco Solitons are known to start out as Rankine vortices of positive Gauss
curvature. They are rotational structures, hence it is conceivable that they
could support complex principal curvatures. Experimentally they decay and
convert into long lived minimal surfaces of negative Gauss curvature. An
explanation for the evolution might be given in terms of a phase change from a
Minimal surface of positive Gauss curvature to a Minimal surface of negative
Gauss curvature.
During the decay process, caustics are observed that have the visual signature
of multiple spiral arms (like the flat spiral galaxies). When the stable Falaco
state appears, the spiral arms have disappeared.
