On highly degenerate CR maps of spheres

Abstract

For $N\ge 4$ we classify the $(N-3)$-degenerate smooth CR maps of the three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is of degree at most two, or equivalent to one of the four maps into the five-dimensional sphere classified by Faran. As a byproduct of our classification we obtain new examples of rational maps of degree three which are $(N-3)$-degenerate only along a proper real subvariety and are not equivalent to polynomial maps. In particular, by changing the base point, it is possible to construct new families of nondegenerate maps.

References in corpus (11)

• Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics
• Complexity results for CR mappings between spheres
• Homotopy equivalence for proper holomorphic mappings
• Hermitian symmetric polynomials and CR complexity
• Classification of Holomorphic Mappings of Hyperquadrics from ${\mathbb{C}}^2$ to ${\mathbb{C}}^3$
• Rational sphere maps, linear programming, and compressed sensing
• Local and infinitesimal rigidity of hypersurface embeddings
• The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps
• Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations
• Deformations of CR maps and applications
• The reflection map and infinitesimal deformations of sphere mappings