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- W4297914509 abstract "In this paper, we study degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $Msubset bC^{n+1}$ into a sphere $bS$ in a higher dimensional complex space $bC^{N+1}$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings $f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank $d$ of the second fundamental form and all of its covariant derivatives is $<n$ (here, $n$ is the CR dimension of $M$), then $f(M)$ is contained in a complex plane of dimension $n+d+1$. The converse of this statement is also true, as is easy to see. When the rank $d$ exceeds $n$, it is no longer true, in general, that $f(M)$ is contained in a complex plane of dimension $n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension $n$, then partial rigidity may still persist, but there is a defect $k$ that arises from the ranks exceeding $n$ such that $f(M)$ is only contained in a complex plane of dimension $n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples." @default.
- W4297914509 created "2022-10-01" @default.
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- W4297914509 date "2012-08-13" @default.
- W4297914509 modified "2023-09-27" @default.
- W4297914509 title "Partial rigidity of degenerate CR embeddings into spheres" @default.
- W4297914509 doi "https://doi.org/10.48550/arxiv.1208.2731" @default.
- W4297914509 hasPublicationYear "2012" @default.
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