Mathematische Annalen

, Volume 358, Issue 3–4, pp 1059–1089

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

Article

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Department of MathematicsOklahoma State UniversityStillwaterUSA
  4. 4.Department of MathematicsPurdue UniversityWest LafayetteUSA

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