In M^{8}-H duality surface with M^{4} projections smaller than four appear as singularities of algebraic surfaces in M^8. The dimension of M^{4} projection varies and known extremals can be interpreted in terms of singularities.

An especially interesting singularity would be a static 3-D singularity M^{1}× X^{2} with a geodesic circle S^{1} ⊂ CP_{2} as a local blow-up.

- The simplest guess is as product M
^{1}× S^{2}× S^{1}. 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 M
^{4}and CP_{2}degrees of freedom is required. In order to obtain a minimal surface, one must achieve a situation in which the S^{2}part of the second fundamental form contains a contribution from a geodesic circle S^{1}⊂ CP_{2}so that its trace vanishes. A simple example would correspond to a soap bubble-like minimal surface with M^{4}projection M^{1}× X^{2}, which has having geodesic circle S^{1}as a local CP_{2}projection, which depends on the point of M^{1}× X^{2}. - The simplest candidate for the minimal surface M
^{1}× S^{2}⊂ M^{4}. One could assign a geodesic circle S^{1}⊂ CP_{2}to each point of S^{2}in such a manner that the orientation of S^{1}⊂ CP_{2}depends on the point of S^{2}. - A natural simplifying assumption is that one has S
^{1}⊂ S^{2}_{1}⊂ CP_{2}, where S^{2}_{1}is a geodesic sphere of CP_{2}which can be either homologically trivial or non-trivial. One would have a map S^{2}→ S^{2}_{1}such that the image point of point of S^{2}defines the position of the North pole of S^{2}_{1}defining the corresponding geodesic circle as the equatorial circle.The maps S

^{2}→ S^{2}_{1}are characterized by a winding number. The map could also depend on the time coordinate for M^{1}so that the circle S^{1}associated with a given point of S^{1}would rotate in S^{2}_{1}. North pole of S^{2}_{1}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 M^{1}so that the circle S^{1}associated with a given point of S^{1}would rotate in S^{2}_{1}.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.