Partial rigidity of degenerate CR embeddings into spheres
Partial rigidity of degenerate CR embeddings into spheres
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DOI：
10.1016/j.aim.2013.02.011
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发表时间：
2012-08
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通讯作者：
P. Ebenfelt
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作者：
P. Ebenfelt
本文研究了高维复空间Cn+1中严格伪凸超曲面M⊂Cn+1到球面S的退化CR嵌入f.利用CR第二基本形式及其协变导数的秩刻划了映射f的退化性质.2004年，作者与X.Huang和D.Zaitsev一起建立了低余维球上CR嵌入f的刚性结果。证明这一结果的关键一步是证明退化映射必然包含在目标球面的复平面截面(部分刚性)中。在2004年的文章中，证明了如果第二基本形式及其所有协变导数的总秩d是<；n(这里n是M的CR维)，则f(M)包含在n+d+1维的复平面上。这一结论也是成立的，这是显而易见的。当总的秩d超过n时，一般而言，f(M)包含在n+d+1维的复平面中不再成立，如通过例子可以看到的。本文系统地研究了球面上的退化CR映射。证明了当第二基本形式及其协变导数的阶数超过CR维n时，部分刚性仍然存在，但存在一个“亏”k，使得f(M)只包含在n+d+k+1维的复平面上，并举例说明了这种“亏”.
In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M⊂Cn+1into a sphere S in a higher dimensional complex space CN+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 in 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 total 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.