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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 monopolepoints. We investigate the limiting behavior of various constant scalarcurvature 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 closeconnection to the orbifold Yamabe problem, which we show is not always solvable(in contrast to the case of compact manifolds). In particular, we show thatthere is no constant scalar curvature orbifold metric in the conformal class ofa conformally compactified non-flat hyperkahler ALE space in dimension four.

Monopole metrics and the orbifold Yamabe problem