Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
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雄心几何 I：爱因斯坦度量和极值 ambikaehler 结构
DOI：
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发表时间：
2013
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通讯作者：
P. Gauduchon
中科院分区：
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作者：
V. Apostolov;D. Calderbank;P. Gauduchon
我们给出了共形等价但定向相反的四维Kaehler度量的局部分类，这些度量相对于普通的2-环面作用是环面的。在一般情况下，根据二次多项式Q和一个变量的任意函数A和B，这些“参照式”结构有一个有趣的局部几何。
我们使用这种描述来分类爱因斯坦4度规，它们在两个方向上都是厄米特的，以及一类爱因斯坦-麦克斯韦方程的解，包括Plebanski-Demianski度规的Riemannian类似。我们的分类可以看作是R.Debever，N.Kamran和R.McLenaghan在相对论中的一个结果的黎曼类比，并且是由R.Bryant和第一和第三作者独立获得的自对偶爱因斯坦厄米特4-流形分类的自然扩展。
这些爱因斯坦度规正是具有消失的巴赫张量的无序结构，因此具有相关的Toric Kaehler度规是极值(在E.Calabi意义下)的性质。我们的主要结果也对后者进行了分类，提供了显式极值Kaehler度量的新例子。对于爱因斯坦-麦克斯韦结构和极值相容结构，A和B都是四次多项式，但对系数的条件不同。在本文的续篇中，我们考虑整体例子，并利用它们来解决具有第二Betti数b2=2的环状4-orbilold上的极值Kaehler度量的存在性问题。
We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable.
We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors.
These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.