球上自动度量衡
Metriques autoduales sur la boule
摘要：
一个4球上的共形度规在边界3球上导出一个共形度规和一个无迹的第二基本型。相反，这样一个3-球面上的数据是一个唯一的自对偶共形度量的边界，定义在球面的邻域中。本文刻画了3-球（接近标准圆度量）上的共形度量和无迹第二基本型，它们是整个4-球上的自对偶共形度量的边界.当边界上的数据被简化为一个共形度规时（第二基本型的无迹部分消失了），人们可能希望在填充度规的共形类中找到一个爱因斯坦度规，在边界上有一个2阶极点。我们确定的3球上的共形度量的边界，这样的自对偶爱因斯坦度量的4球。特别是，这意味着LeBrun的正频率猜想。证明使用扭量理论，这使得翻译的问题，在复分析，这使我们证明了一个标准，某些可积的CR结构的签名（1，1）是可填充的复域。最后，我们解决了一个类似的高维问题：用四元数Kahler度规代替自对偶Einstein度规，用四元数切触结构代替边界上的共形结构（作者以前介绍过）;与四维情况相反，证明了（4 m-1）-球面上标准四元数切触结构的任何小变形都是四元数-4米球上的卡勒度量。
英文摘要：
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-Kahler 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-Kahler metric on the 4m-ball.
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