The Hopf Map and Minimal surfaces of imaginary principal curvatures

Application of generalized Implicit Hypersurface theory indicates that the Hopf map generates a 2-D 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 2-D surface.)
Hence the Adjoint curvature is zero, the Gauss curvature is positive and real, while the mean curvature is zero.

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.


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Last update 01/23/2009
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