Geometry & Topology

Toric LeBrun metrics and Joyce metrics

Nobuhiro Honda and Jeff Viaclovsky

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We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

Article information

Geom. Topol., Volume 17, Number 5 (2013), 2923-2934.

Received: 9 August 2012
Accepted: 31 May 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry

toric self-dual metrics


Honda, Nobuhiro; Viaclovsky, Jeff. Toric LeBrun metrics and Joyce metrics. Geom. Topol. 17 (2013), no. 5, 2923--2934. doi:10.2140/gt.2013.17.2923.

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  • A Fujiki, Compact self-dual manifolds with torus actions, J. Differential Geom. 55 (2000) 229–324
  • G W Gibbons, S W Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978) 430–432
  • N Honda, J Viaclovsky, Conformal symmetries of self-dual hyperbolic monopole metrics, Osaka J. Math. 50 (2013) 197–249
  • D D Joyce, Explicit construction of self-dual $4$–manifolds, Duke Math. J. 77 (1995) 519–552
  • C LeBrun, Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann. 289 (1991) 383–392
  • C LeBrun, Explicit self-dual metrics on ${\bf C}{\rm P}_2\,\#\,\cdots\,\#\,{\bf C}{\rm P}_2$, J. Differential Geom. 34 (1991) 223–253
  • C LeBrun, Self-dual manifolds and hyperbolic geometry, from: “Einstein metrics and Yang–Mills connections”, (T Mabuchi, S Mukai, editors), 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