Métriques autoduales sur la boule
Métriques autoduales sur la boule
复制标题
圆球上的自动测量
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Olivier Biquard
中科院分区：
文献类型：
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作者：
Olivier Biquard
摘要。一个四球上的共形度量在边界三球上推导出一个共形度量和一个无迹的第二基本形式。相反，3球上的数据是在球的邻域中定义的唯一自对偶共形度量的边界。本文刻画了3球上的共形度量和无迹第二基本形式（接近标准圆度量），它们是整个4球上的自对偶共形度量的边界。当边界上的数据被约化为一个共形度量（第二基本形式的无迹部分消失）时，人们可能希望在填充度量的共形类中找到一个爱因斯坦度量，在边界上具有2阶的极点。我们确定了3球上的哪些共形度量是4球上的自对偶爱因斯坦度量的边界。特别地，这隐含了LeBrun的正频率猜想。证明中使用了扭转理论，可以将问题转化为复数分析；这使得我们证明了签名（1,1）的某些可积CR结构可被复域填充的一个准则。最后，我们解决了一个类似的高维问题：自对偶爱因斯坦度量被quaternionic-Kähler度量取代，边界上的共形结构被四元数接触结构（作者之前介绍过）取代；与四维情况相比，我们证明了（4m−1）球上标准四元数接触结构的任何小变形都是（4m）球上quaternionic-Kähler度规的边界。
Abstract.A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-Kähler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-Kähler metric on the (4m)-ball.