Bubble-like structures in TGD Universe

Membrane-like structures appear in all length scales from soap bubbles to large cosmic voids and it would be nice if they were fundamental objects in the TGD Universe. The Fermi bubble in the galactic center is an especially interesting membrane-like structure also from the TGD point of view as also the membrane-like structure presumably associated with the analog of horizon of the TGD counterpart of blackhole. Cell membrane is an example of a biological structure of this kind. I have however failed to identify candidates for the membrane-like structures.

In M⁸-H duality surface with M⁴ projections smaller than four appear as singularities of algebraic surfaces in M^8. The dimension of M⁴ projection varies and known extremals can be interpreted in terms of singularities.

An especially interesting singularity would be a static 3-D singularity M¹× X² with a geodesic circle S¹ ⊂ CP₂ as a local blow-up.

1. The simplest guess is as product M¹× S²× S¹. The problem is that a soap bubble is not a minimal surface: a pressure difference between interior and exterior of the bubble is required so that the trace of the second fundamental form is constant. Quite generally, closed 2-D surfaces cannot be minimal surfaces in a flat 3-space since the vanishing curvature of the minimal surface forces the local saddle structure.
2. A correlation between M⁴ and CP₂ degrees of freedom is required. In order to obtain a minimal surface, one must achieve a situation in which the S² part of the second fundamental form contains a contribution from a geodesic circle S¹ ⊂ CP₂ so that its trace vanishes. A simple example would correspond to a soap bubble-like minimal surface with M⁴ projection M¹× X², which has having geodesic circle S¹ as a local CP₂ projection, which depends on the point of M¹× X².
3. The simplest candidate for the minimal surface M¹× S²⊂ M⁴. One could assign a geodesic circle S¹⊂ CP₂ to each point of S² in such a manner that the orientation of S¹⊂ CP₂ depends on the point of S².

4. A natural simplifying assumption is that one has S¹⊂ S²₁⊂ CP₂, where S²₁ is a geodesic sphere of CP₂ which can be either homologically trivial or non-trivial. One would have a map S²→ S²₁ such that the image point of point of S² defines the position of the North pole of S²₁ defining the corresponding geodesic circle as the equatorial circle.

   The maps S²→ S²₁ are characterized by a winding number. The map could also depend on the time coordinate for M¹ so that the circle S¹ associated with a given point of S¹ would rotate in S²₁. North pole of S²₁ defining the corresponding geodesic circle as an equatorial circle. These maps are characterized by a winding number. The map could also depend on the time coordinate for M¹ so that the circle S¹ associated with a given point of S¹ would rotate in S²₁.

   The minimal surface property might be realized for maximally symmetric maps. Isometric identification using map with winding number n=+/- 1 is certainly the simplest imaginable possibility.

See What could 2-D minimal surfaces teach about TGD? or the chapter Zero Energy Ontology.

For a summary of earlier postings see Latest progress in TGD.

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