Monopole metrics and the orbifold Yamabe problem

Viaclovsky, Jeff A.

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2503-2543.

Nous considérons les classes conformes auto-duales sur $n\#{\mathrm{ℂ\mathbb{P}}}^2$ introduites par LeBrun. Elles dépendent du choix de $n$ points dans l’espace hyperbolique de dimension 3, appelés points de monopôle. Nous étudions les limites de diverses métriques de courbure scalaire constante dans ces classes conformes lorsque ces points se rapprochent ou tendent vers le bord de l’espace hyperbolique. Il existe une relation étroite avec le problème de Yamabe sur les orbifolds qui n’admet pas toujours de solution (contrairement au cas des variétés compactes). En particulier, nous montrons qu’il n’existe pas de métrique d’orbifold de courbure scalaire constante dans la classe conforme d’un espace ALE hyperkählérien conformément compact en dimension 4.

We consider the self-dual conformal classes on $n\#{\mathrm{ℂ\mathbb{P}}}^2$ 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 hyperkähler ALE space in dimension four.

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