Monopole metrics and the orbifold Yamabe problem

Details Monopole metrics and the orbifold Yamabe problem Monopole metrics and the orbifold Yamabe problem

2010-02-10

arxiv.org/abs/1002.2119v2

Mathematics

Differential Geometry

34 pages, to appear in Annales de L'Institut Fourier

Scientific paper

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.

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