Monopole metrics and the orbifold Yamabe problem
Monopole metrics and the orbifold Yamabe problem
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单极子度量和 Orbifold Yamabe 问题
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发表时间：
2010
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通讯作者：
Jeff A. Viaclovsky
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作者：
Jeff A. Viaclovsky
我们考虑 LeBrun 发现的 n#CP^2 上的自对偶共形类。这些取决于双曲 3 空间中 n 个点（称为单极点）的选择。我们研究了这些共角类中各种常量标量曲率度量在点彼此接近或点趋向双曲空间边界时的限制行为。与 Orbifold Yamabe 问题有密切的联系，我们证明该问题并不总是可解的（与紧凑流形的情况相反）。特别是，我们证明在四维共形紧致非平坦超卡勒 ALE 空间的共角类中不存在恒定标量曲率轨道度量。
We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkahler ALE space in dimension four.