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

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## 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.

## Notes

### Acknowledgments

The authors would like to acknowledge John D’Angelo for many useful conversations on the subject and suggestions related to this project. The second author would also like to thank Peter Ebenfelt for useful discussions on this subject. The authors are greatly indebted to the referee who pointed out several errors in the presentation of the results and helped uncover a gap in the proof of Lemma 11, which has been fixed. Finally the authors would like to acknowledge MSRI and AIM for holding workshops on the subject of CR complexity, which the authors attended and which led to the present project.

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