Conformal symmetries of self-dual hyperbolic monopole metrics

We determine the group of conformal automorphisms of the self-dual metrics on n#CP² due to LeBrun for n≥3, and Poon for n=2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H³ minus a finite number of points, called monopole points. We show that for n≥3, any conformal automorphism is a lift of an isometry of H³ 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 (U(1)×U(1))×D₄, where D₄ is the dihedral group of order 8.