Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

@article{Grundmeier2014BoundingTR,
  title={Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics},
  author={Dusty E. Grundmeier and Jiř{\'i} Lebl and Liz Raquel Vivas},
  journal={Mathematische Annalen},
  year={2014},
  volume={358},
  pages={1059-1089}
}
Using Green’s hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi–Huang and Baouendi–Ebenfelt–Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric… 
An Application of Macaulay's Estimate to CR Geometry
Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials $r(z,\bar{z})$ on $\C^n \times \C^n$ for which $r(z,\bar{z})\norm{z}^{2d}=\norm{h(z)}^2$ for some natural
An application of Macaulay’s estimate to sums of squares problems in several complex variables
Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials r(z, z) on Cn×Cn for which r(z, z) ‖z‖ = ‖h(z)‖ for some natural number d and a holomorphic polynomial
A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture
We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for OPn(d). Our theorem allows us
Initial monomial invariants of holomorphic maps
We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find
Invariant CR Mappings between Hyperquadrics
We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to D'Angelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric
On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials
We show that the Huang-Ji-Yin (HJY) Gap Conjecture concerning CR mappings between spheres follows from a conjecture regarding Sums of Squares (SOS) of polynomials. The connection between the two
Sum of squares conjecture: the monomial case in $$\mathbb {C}^3$$
The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $$r(z,\bar{z})$$ on $$\mathbb {C}^3$$ with diagonal coefficient matrix. This conjecture describes the
Algebraic properties of Hermitian sums of squares
ABSTRACT We study the class of bihomogeneous polynomials on for which there is a positive integer d such that can be written as a Hermitian sum of squares. We reinterpret this problem in terms of
CR Complexity and Hyperquadric Maps
We survey aspects of CR complexity for maps between spheres and hyperquadrics, provide some new interpretations of the maps found by Lebl and Reiter, and indicate how group-invariance fit into the
...
1
2
...

References

SHOWING 1-10 OF 18 REFERENCES
Holomorphic mappings between hyperquadrics with small signature difference
In this paper, we study holomorphic mappings sending a hyperquadric of signature $\ell$ in ${\Bbb C}^n$ into a hyperquadric of signature $\ell'$ in ${\Bbb C}^N$. We show (Theorem 1.1) that if the
Extending proper holomorphic mappings of positive codimension
In this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real-analytic boundaries in complex spaces of different dimensions.
Positivity conditions for Hermitian symmetric functions
We introduce a countable collection of positivity classes for Hermitian symmetric functions on a complex manifold, and establish their basic properties. We study a related notion of stability. The
Embedding strictly pseudoconvex domains into balls
Every relatively compact strictly pseudocc)nvex domain D with C2 boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n > 2
A new gap phenomenon for proper holomorphic mappings from B^n into B^N
In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then
Hermitian Symmetric Polynomials and CR Complexity
Properties of Hermitian forms are used to investigate several natural questions from CR geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs
COMPLEXITY RESULTS FOR CR MAPPINGS BETWEEN SPHERES
Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to
Super-rigidity for holomorphic mappings between hyperquadrics with positive signature
We study local holomorphic mappings sending a piece of a real hyperquadric in a complex space into a hyperquadric in another complex space of possibly larger dimension. We show that these mappings
Hermitian analogues of Hilbert's 17-th problem
...
1
2
...