The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4-D space-times with hyperbolic metric provide canonical representation for a large class of four-manifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals.
4-D hyperbolic space with Minkowski signature is locally isometric with AdS4. This suggests a connection with AdS4/CFT correspondence of M-theory. The boundary of AdS would be now replaced with 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of complex coordinate at light-like boundary. AdS would presumably represent a special case of a more general family of space-time surfaces with constant Ricci scalar satisfying Einstein-Maxwell equations and generalizing the AdS4/CFT correspondence.
For the ordinary AdS5 correspondence empty M4 is identified as boundary. In the recent case the boundary of AdS4 is replaced with a 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. String world sheets have boundaries along light-like 3-surfaces and space-like 3-surfaces at the light-like boundaries of CD. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of hyper- complex coordinate at light-like 3-surface. AdS5× S5 of M-theory context is replaced by a 4-surface of constant Ricci scalar in 8-D imbedding space M4× CP2 satisfying Einstein-Maxwell equations. A generalization of AdS4/CFT correspondence would be in question. There is however a strong objection from cosmology: the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS4 instead of AdS4.
These observations give motivations for finding whether AdS4 and/or AdS4 allows an imbedding as vacuum extremal to M4× S2⊂ M4× CP2, where S2 is a homologically trivial geodesic sphere of CP2. It is easy to guess the general form of the imbedding by writing the line elements of, M4, S2, and AdS4.
The conclusion is that AdS4 allows a local imbedding as a vacuum extremal. Whether also an imbedding as a non-vacuum preferred extremal to homologically non-trivial geodesic sphere is possible, is an interesting question. The only modification in the case of De Sitter space dS4 is the replacement of the function A= 1+y2 appearing in the metric of AdS4 with A=1-y2. Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. p-Adic length scale hypothesis motivates the conjecture that stars indeed have onion-like layered structure consisting of shells, whose radii are consistent with p-adic length scale hypothesis. This brings in mind also Titius-Bode law.
For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".