Tohoku Mathematical Journal

Einstein metrics and Yamabe invariants of weighted projective spaces

Jeff A. Viaclovsky

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An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.

Article information

Tohoku Math. J. (2), Volume 65, Number 2 (2013), 297-311.

First available in Project Euclid: 25 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]

Weighted projective spaces Einstein metrics Kähler-Einstein metrics orbifold Yamabe invariants


Viaclovsky, Jeff A. Einstein metrics and Yamabe invariants of weighted projective spaces. Tohoku Math. J. (2) 65 (2013), no. 2, 297--311. doi:10.2748/tmj/1372182728.

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  • K. Akutagawa and B. Botvinnik, Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal. 13 (2003), 259–333.
  • K. Akutagawa and B. Botvinnik, The Yamabe invariants of orbifolds and cylindrical manifolds, and $L^2$-harmonic spinors, J. Reine Angew. Math. 574 (2004), 121–146.
  • K. Akutagawa, Computations of the orbifold Yamabe invariant, Math. Z. 271 (2012), 611–625.
  • M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.
  • R. L. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), 623–715.
  • A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405–483.
  • L. David and P. Gauduchon, The Bochner-flat geometry of weighted projective spaces, Perspectives in Riemannian geometry, 109–156, CRM Proc. Lecture Notes 40, Amer. Math. Soc., Providence, RI, 2006.
  • T. Eguchi and A. J. Hanson, Self-dual solutions to Euclidean gravity, Ann. Physics 120 (1979), 82–106.
  • Y. Fukumoto, The index of the ${\rm Spin}^c$ Dirac operator on the weighted projective space and the reciprocity law of the Fourier-Dedekind sum, J. Math. Anal. Appl. 309 (2005), 674–685.
  • M. J. Gursky and C. LeBrun, Yamabe invariants and ${\rm Spin}^c$ structures, Geom. Funct. Anal. 8 (1998), 965–977.
  • J. P. Gauntlett, D. Martelli, J. Sparks, and S.-T. Yau, Obstructions to the existence of Sasaki-Einstein metrics, Comm. Math. Phys. 273 (2007), 803–827.
  • N. J. Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geom. 9 (1974), 435–441.
  • N. J. Hitchin, Einstein metrics and the eta-invariant, Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl. 95–105.
  • F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Mathematics Lecture Series, No. 3, Publish or Perish Inc., Boston, Mass. 1974.
  • T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206 (1973), 243–248.
  • W. Kühnel, Conformal transformations between Einstein spaces, Conformal geometry (Bonn, 1985/1986), 105–146, Aspects Math. E12, Vieweg, Braunschweig, 1988.
  • C. LeBrun, Einstein metrics and the Yamabe problem, Trends in mathematical physics (Knoxville, TN, 1998), 353–376, AMS/IP Stud. Adv. Math. vol. 13, Amer. Math. Soc. Providence, RI, 1999.
  • H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Math. Ser. 38, Princeton University Press, Princeton, NJ, 1989.
  • M. T. Lock and J. A. Viaclovsky, Anti-self-dual orbifolds with cyclic quotient singularities, arXiv:1205.4059, 2012.
  • T. Mabuchi, Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties, Osaka J. Math. 24 (1987), 705–737.
  • A. Moroianu, Parallel and Killing spinors on ${\rm Spin}^c$ manifolds, Comm. Math. Phys. 187 (1997), 417–427.
  • H. Nakajima, Self-duality of ALE Ricci-flat 4-manifolds and positive mass theorem, Recent topics in differential and analytic geometry, 385–396, Academic Press, Boston, MA, 1990.
  • M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971/72), 247–258.
  • H. Rademacher, Generalization of the reciprocity formula for Dedekind sums, Duke Math. J. 21 (1954), 391–397.
  • J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 (2011), 109–159.
  • R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479–495.
  • Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275.
  • C. W. Tønnesen-Friedman, Extremal Kähler metrics and Hamiltonian functions. II, Glasg. Math. J. 44 (2002), 241–253.
  • J. A. Thorpe, Some remarks on the Gauss-Bonnet integral, J. Math. Mech. 18 (1969), 779–786.
  • G. Tian and J. Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005), 346–372.
  • J. Viaclovsky, Monopole metrics and the orbifold Yamabe problem, Annales de L'Institut Fourier 60 (2010), 2503–2543.
  • D. B. Zagier, Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the $G$-signature theorem to transformation groups, symmetric products and number theory, Lecture Notes in Mathematics, Vol. 290, Springer-Verlag, Berlin, 1972.