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

Abstract

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any hyperquadric in $\C^3$, resulting in a finite list of equivalence classes. Second, we prove that all degree-two CR maps of spheres in all dimensions are spherically equivalent to a monomial map, thus obtaining an elegant classification of all degree-two CR sphere maps.

23 pages; accepted to Michigan Math. J

References in corpus (5)

• Canonical forms for complex matrix congruence and *congruence
• Complexity results for CR mappings between spheres
• On the complexity of proper holomorphic mappings between balls
• A new gap phenomenon for proper holomorphic mappings from B^n into B^N
• Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

Cited by in corpus (16)

• Homotopy equivalence for proper holomorphic mappings
• Hermitian symmetric polynomials and CR complexity
• Classification of Holomorphic Mappings of Hyperquadrics from ${\mathbb{C}}^2$ to ${\mathbb{C}}^3$
• Pfister's theorem fails in the Hermitian case
• Rational sphere maps, linear programming, and compressed sensing
• Local and infinitesimal rigidity of hypersurface embeddings
• Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations
• Exhaustion functions and normal forms for proper maps of balls
• Initial monomial invariants of holomorphic maps
• Topological Aspects of Holomorphic Mappings of Hyperquadrics from ${\mathbb{C}}^2$ to ${\mathbb{C}}^3$
• Polynomials constant on a hyperplane and CR maps of hyperquadrics
• D'Angelo conjecture in the third gap interval
• Rational Maps of Balls and their Associated Groups
• On highly degenerate CR maps of spheres
• Polynomials constant on a hyperplane and CR maps of spheres
• Symmetries in CR complexity theory