Mathematics – Complex Variables
Scientific paper
2011-10-18
Mathematics
Complex Variables
22 pages, 1 figure, obsoletes arXiv:1006.0186
Scientific paper
Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials can be 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. If we have a real-analytic CR mapping of a hyperquadric not equivalent to a sphere to another hyperquadric $Q(A,B)$, then 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.
Grundmeier Dusty
Lebl Jiri
Vivas Liz
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