多胞形度量的分类和具有两个对称的完备标量平坦Kähler 4-流形
Classification of polytope metrics and complete scalar-flat Kähler 4-Manifolds with two symmetries
摘要：
我们研究无界的二维度量多面体，如那些出现的Kähler的完全Kähler 4-流形与两个交换对称性和零数量曲率的Kähler替代。在弱闭性条件下，我们得到了这类多面体上度量的完全分类，并由此对相应Kähler 4-流形上所有可能的度量进行了分类.如果多面体是平面或半平面，那么只有平坦的度量是可能的，如果多面体有一个角，那么2-参数族的广义Taub-NUT（由唐纳森发现）确实是唯一可能的度量。具有$n\ge3$边的多面体允许可能度量的$(n+2)$维族。
英文摘要：
We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible metrics on on the corresponding Kähler 4-manifolds. If the polytope is the plane or half-plane then only flat metrics are possible, and if the polytope has one corner then the 2-parameter family of generalized Taub-NUTs (discovered by Donaldson) are indeed the only possible metrics. Polytopes with $n\ge3$ edges admit an $(n+2)$-dimensional family of possible metrics.
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