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

TL;DR

This paper establishes bounds on the rank of Hermitian forms using hyperplane restriction theorems and applies these results to prove rigidity and stability theorems for CR mappings between hyperquadrics, including infinite-dimensional cases.

Contribution

It introduces a new bound on Hermitian form ranks and derives rigidity and stability results for CR mappings between hyperquadrics, extending to infinite-dimensional settings.

Findings

Rank of Hermitian forms is bounded by a constant depending on affine restrictions.

CR mappings from hyperquadrics are either contained in a complex affine subspace or have bounded parameters.

Existence of nontrivial CR mappings when parameters are large and comparable.

Abstract

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 $Q(A,B)$, either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.