Osaka Journal of Mathematics

Conformal symmetries of self-dual hyperbolic monopole metrics

Nobuhiro Honda and Jeff Viaclovsky

Full-text: Open access


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

Permanent link to this document

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.

Export citation


  • M.F. Atiyah, N.J. Hitchin and I.M. Singer: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425–461.
  • A.L. Besse: Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10, Springer, Berlin, 1987.
  • A. Fujiki: Compact self-dual manifolds with torus actions, J. Differential Geom. 55 (2000), 229–324.
  • A. Fujiki: Automorphism groups of Joyce twistor spaces, preprint (2012).
  • G.W. Gibbons and C.M. Warnick: Hidden symmetry of hyperbolic monopole motion, J. Geom. Phys. 57 (2007), 2286–2315.
  • N.J. Hitchin: Complex manifolds and Einstein's equations; in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math. 970, Springer, Berlin, 1992, 73–99.
  • N. Honda and J. Viaclovsky: Toric LeBrun metrics and Joyce metrics, arXiv:1208.2065.
  • D.D. Joyce: Explicit construction of self-dual $4$-manifolds, Duke Math. J. 77 (1995), 519–552.
  • P.E. Jones and K.P. Tod: Minitwistor spaces and Einstein–Weyl spaces, Classical Quantum Gravity 2 (1985), 565–577.
  • B. Kreussler and H. Kurke: Twistor spaces over the connected sum of 3 projective planes, Compositio Math. 82 (1992), 25–55.
  • J. Kollár: The structure of algebraic threefolds: an introduction to Mori's program, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 211–273.
  • \begingroup C. LeBrun: Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann. 289 (1991), 383–392. \endgroup
  • C. LeBrun: Explicit self-dual metrics on $\mathbf{C}\mathrm{P}_{2} \# \cdots \# \mathbf{C}\mathrm{P}_{2}$, J. Differential Geom. 34 (1991), 223–253.
  • C. LeBrun: Self-dual manifolds and hyperbolic geometry; in Einstein Metrics and Yang–Mills Connections (Sanda, 1990), Lecture Notes in Pure and Appl. Math. 145, Dekker, New York, 1993, 99–131.
  • Y.S. Poon: Compact self-dual manifolds with positive scalar curvature, J. Differential Geom. 24 (1986), 97–132.
  • Y.S. Poon: On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), 451–491.
  • Y.S. Poon: Conformal transformations of compact self-dual manifolds, Internat. J. Math. 5 (1994), 125–140.
  • H. Pedersen and Y.S. Poon: Equivariant connected sums of compact self-dual manifolds, Math. Ann. 301 (1995), 717–749.
  • J.G. Ratcliffe: Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer, New York, 1994.