Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics
Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics
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DOI：
10.1007/s00208-013-0989-z
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发表时间：
2011-10
影响因子：
1.4
通讯作者：
Dusty Grundmeier;Jiří Lebl;L. Vivas
中科院分区：
文献类型：
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作者：
Dusty Grundmeier;Jiří Lebl;L. Vivas
利用绿色的超平面限制定理，证明了全纯多项式空间上的埃尔米特形式的秩是有界的，这个常数只依赖于该形式限制在仿射流形上的最大秩.作为应用，我们在Baouendi-Huang和Baouendi-Ebenfelt-Huang结果的基础上证明了超二次曲面间CR映射的一个刚性定理。给定一个超二次曲面（不等价于球面）到另一个超二次曲面的实解析CR映射，该映射的象要么包含在复仿射子空间中，要么被一个只依赖于的常数所约束.最后，我们证明了超二次曲面的非平凡CR映射存在的一个稳定性结果.也就是说，只要和都足够大且可比较，则存在其像不包含在超平面中的CR映射。当映射到无限维希尔伯特空间中的超二次曲面时，刚性结果也得到了推广。
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, either the image of the mapping is contained in a complex affine subspace, oris bounded by a constant depending only on. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as bothandare 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.