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

- 139 Downloads
- 1 Citations

## 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.

## References

- 1.Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York (1993) MR 1255973Google Scholar
- 2.Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton (1999) MR 1668103Google Scholar
- 3.Baouendi, M.S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. Am. J. Math.
**133**(6), 1633–1661 (2011). doi: 10.1353/ajm.2011.0044. MR 2863372Google Scholar - 4.Baouendi, M.S., Huang, X.: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Differ. Geom.
**69**(2):379–398 (2005) MR 2169869Google Scholar - 5.D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics, CRC Press, Boca Raton (1993) MR 1224231Google Scholar
- 6.D’Angelo, J.P., Lebl, J.: Complexity results for CR mappings between spheres. Internat. J. Math.
**20**(2):149–166 (2009) MR 2493357Google Scholar - 7.D’Angelo, J.P., Lebl, J.: Hermitian symmetric polynomials and CR complexity. J. Geom. Anal.
**21**(3):599–619 (2011) MR 2810845Google Scholar - 8.D’Angelo, J.P.: Hermitian analogues of Hilbert’s 17-th problem. Adv. Math.
**226**(5):4607–4637 (2011) MR 2770459Google Scholar - 9.D’Angelo, J.P.: Dror Varolin, positivity conditions for Hermitian symmetric functions. Asian J. Math.
**8**(2):215–231 (2004) MR 2129535Google Scholar - 10.Forstnerič, F.: Extending proper holomorphic mappings of positive codimension. Invent. Math.
**95**(1):31–61 (1989) MR 969413Google Scholar - 11.Forstnerič, F.: Embedding strictly pseudoconvex domains into balls. Trans. Am. Math. Soc.
**295**(1), 347–368 (1986). doi: 10.2307/2000160. MR 831203Google Scholar - 12.Green, M.: Generic Initial Ideals, Six Lectures on Commutative Algebra (Bellaterra, 1996). Programming in Mathematics, vol. 166, pp. 119–186. Birkhäuser, Basel (1998) MR 1648665Google Scholar
- 13.Green, M.: Restrictions of Linear Series to Hyperplanes, and Some Results of Macaulay and Gotzmann, Algebraic Curves and Projective Geometry, pp. 76–86. Springer, Berlin (1989) MR 1023391Google Scholar
- 14.Huang, X., Ji, S., Xu, D.: A new gap phenomenon for proper holomorphic mappings from \(B^n\) into \(B^N\). Math. Res. Lett.
**13**(4):515–529 (2006) MR 2250487Google Scholar - 15.Lempert, L.: Imbedding Cauchy-Riemann manifolds into a sphere. Internat. J. Math.
**1**(1), 91–108 (1990). doi: 10.1142/S0129167X90000071. MR 1044662Google Scholar - 16.Lewy, H.: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables. Ann. Math.(2),
**64**, 514–522 (1956) MR 0081952Google Scholar