Universal Models For Real Submanifolds

In previous papers by the present author, a machinery for calculating automorphisms, constructing invariants, and classifying real submanifolds of a complex manifold was developed. The main step in this machinery is the construction of a “nice” model surface. The nice model surface can be treated as an analog of the osculating paraboloid in classical differential geometry. Model surfaces suggested earlier possess a complete list of the desired properties only if some upper estimate for the codimension of the submanifold is satisfied. If this estimate fails, then the surfaces lose the universality property (that is, the ability to touch any germ in an appropriate way), which restricts their applicability. In the present paper, we get rid of this restriction: for an arbitrary type (n,K) (where n is the dimension of the complex tangent plane, and K is the real codimension), we construct a nice model surface. In particular, we solve the problem of constructing a nondegenerate germ of a real analytic submanifold of a complex manifold of arbitrary given type (n,K) with the richest possible group of holomorphic automorphisms in the given class.     

The normal form of germs of four-dimensional real submanifolds in ℂ n at generic ℝℂ-singular points

Mathematical Notes -     

Precision magnetic quadrupole lens for a nuclear scanning microprobe based on an ÉGP-10 electrostatic tandem accelerator

Basic ideas underlying the design of a precision magnetic quadrupole lens for a nuclear scanning microprobe with a maximal accelerating voltage of 14 MV are set forth. Four magnetic quadrupoles are combined into doublets. The doublets are placed on adjusting gears, which bring the local coordinate system of each lens into coincidence with the laboratory system related to the axis of the beam. Each lens provides a maximal gradient of the field of 0.68 T/cm, which makes it possible to perform stigmatic focusing of the beam with a working distance of 22 cm. All lenses are nonseparable and made of one piece of high-quality electrical steel. A special lens-feeding unit is designed that provides manual and remote control of pole tip excitation.     

Stabilizer of a function in the gage group

It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) ? x of complexity two, (4) d = 0 in all remaining cases.     

Homogeneous Hypersurfaces in ℂ<Superscript>3</Superscript>, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ³ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ³ with finite-dimensional infinitesimal automorphism algebra \mathfrakg\mathfrak{g} and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all \mathfrakg\mathfrak{g}-homogeneous hypersurfaces in ℂ³ and also classify the corresponding local transitive actions of the model algebra \mathfrakg\mathfrak{g} on hypersurfaces in ℂ³.