Normal forms, hermitian operators, and CR maps of spheres and hyperquadrics

Topological aspects of holomorphic mappings of hyperquadrics from ℂ2to ℂ3

“…We study holomorphic mappings between the sphere S 2 ⊂ C 2 and the hyperquadric S 3 ε ⊂ C 3 , which for ε = ±1 is given by S 3 ± := (z 1 , z 2 , z 3 ) ∈ C 3 : |z 1 | 2 + |z 2 | 2 ± |z 3 | 2 = 1 , so that S 3 + = S 3 is the sphere in C 3 . Faran [Far82] classified holomorphic mappings between spheres in C 2 and C 3 and Lebl [Leb11] classified mappings sending S 2 to S 3 − . In [Rei14a] we give a new CR-geometric approach to reprove Faran's and Lebl's results in a unified manner.…”

Hermitian Symmetric Polynomials and CR Complexity

“…In the notation of this paper, the main result in [15] implies that D * (H(2; 3, 1) ∩ I (r)) = 3 and the main result in [19] states that D * (H(2; 2, 2) ∩ I (r)) = 3. Combining the results from [14,19] and this section yields an almost complete answer in domain dimension two. We do not know whether D * (J ) is finite in case (A, B) is (3,2) or (2,3).…”

“…Therefore, assuming that in(p) = (2, 1, 0), p has to be the defining equation for a sphere, and pq has the signature for a defining equation of the hyperquadric Q(2, 1). The second author verified in [19] that there is (up to a linear change of coordinates) only one such map, and therefore it must be the one above. Furthermore, pq cannot be made diagonal after any linear change of coordinates.…”

“…The classification in [19], which relies on Faran's work [14], includes finding all elements of H(2; 2, 2) ∩ I (r). In the notation of this paper, the main result in [15] implies that D * (H(2; 3, 1) ∩ I (r)) = 3 and the main result in [19] states that D * (H(2; 2, 2) ∩ I (r)) = 3.…”

“…Analyzing the cases of small rank is difficult. We rely on Faran's classification of planar maps [13] and recent work in [19]. The table after Theorem 7.1 nicely summarizes the situation when n = 2.…”