Osaka Journal of Mathematics

Conformal symmetries of self-dual hyperbolic monopole metrics

Nobuhiro Honda and Jeff Viaclovsky

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We determine the group of conformal automorphisms of the self-dual metrics on $n \# \mathbb{CP}^{2}$ due to LeBrun for $n \geq 3$, and Poon for $n = 2$. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space $\mathcal{H}^{3}$ minus a finite number of points, called monopole points. We show that for $n \geq 3$, any conformal automorphism is a lift of an isometry of $\mathcal{H}^{3}$ which preserves the set of monopole points. Furthermore, we prove that for $n = 2$, such lifts form a subgroup of index $2$ in the full automorphism group, which we show to be a semi-direct product $(\mathrm{U}(1) \times \mathrm{U}(1)) \rtimes \mathrm{D}_{4}$, where $\mathrm{D}_{4}$ is the dihedral group of order $8$.

Article information

Osaka J. Math. Volume 50, Number 1 (2013), 197-249.

First available in Project Euclid: 27 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry
Secondary: 53C28: Twistor methods [See also 32L25]


Honda, Nobuhiro; Viaclovsky, Jeff. Conformal symmetries of self-dual hyperbolic monopole metrics. Osaka J. Math. 50 (2013), no. 1, 197--249.

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