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