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Constant scalar curvature metrics on Hirzebruch surfaces

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An Erratum to this article was published on 20 July 2014

Abstract

We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. The construction is reduced to an ordinary differential equation called the Duffing equation. An ODE for Bach-flat metrics on Hirzebruch surfaces with large isometry group is also derived.

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Notes

  1. In the present paper, a metric always refers to a (positive-definite) Riemannian metric of class \(C^{\infty }\) unless stated otherwise.

  2. Provided \(m\ge 1\), the connection \({\fancyscript{H}}_m\) is not integrable since the associated principal connection \(\widetilde{{\fancyscript{H}}}_m\) corresponds to \(m\check{\omega }\), the area form \(\check{\omega }\) of the Fubini-Study metric on \(\mathbb {CP}^1\) multiplied by \(m\) (cf. [26]).

  3. For its proof, we note that if \(\phi =\phi (t)\) is a function on \(S^3\times (-T, T)\) depending only on \(t\), then \(X(\phi )=Y(\phi )=V(\phi )=0\) and \(\partial _t(\phi )=\phi '(t)\).

  4. At this moment, the author does not know if a Bach-flat metric of the latter case is locally conformally Einstein (cf. [14]).

References

  1. Abbena, E., Garbiero, S., Salamon, S.: Bach-flat Lie groups in dimension \(4\). C. R. Math. Acad. Sci. Paris, Ser. I 351(7–8), 303–306 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, R., O’Regan, D.: An Introduction to Ordinary Differential Equations, Universitext. Springer, New York (2008)

    Book  Google Scholar 

  3. Agarwal, R., O’Regan, D.: Ordinary and Partial Differential Equations, Universitext. Springer, New York (2009)

    Google Scholar 

  4. Bérard-Bergery, L.: Sur de nouvelles variétés riemanniennes d’Einstein. Publications de l’Institut Élie Cartan, Nancy (1982)

    Google Scholar 

  5. Bérard-Bergery, L., Bourguignon, J.-P.: Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math. 26(2), 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  6. Besse, A.L.: Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb, vol. 93. Springer, Berlin (1978)

    Book  Google Scholar 

  7. Besse, A.L.: Géométrie riemannienne en dimension 4, Papers from the Arthur Besse seminar held at the Université de Paris. Paris (1978/1979)

  8. Besse, A.L.: Einstein Manifolds (reprint of the 1987 edition), Classics in Mathematics. Springer, Berlin (2008)

    Google Scholar 

  9. Böhm, C., Wang, M., Ziller, W.: A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal. 14(4), 681–733 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bordoni, M.: Spectra of Submersions. Contemporary Geometry and Related Topics, pp. 51–62. Univ Belgrade Fac Math, Belgrade (2006)

    Google Scholar 

  11. Bowman, F.: Introduction to elliptic functions with applications. English Universities Press Ltd., London (1953)

    MATH  Google Scholar 

  12. Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier (Grenoble) 49(1), vii, xi, 287–302 (1999)

    Google Scholar 

  13. Cheeger, J.: A Lower Bound for the Smallest Eigenvalue of the Laplacian. Problems in Analysis, pp. 195–199. Princeton University Press, Princeton (1970)

    Google Scholar 

  14. Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compos. Math. 49(3), 405–433 (1983)

    MATH  Google Scholar 

  15. Gursky, M.J., Viaclovsky, J.A.: Critical metrics on connected sums of Einstein four-manifolds. arXiv:1303.0827 (2013)

  16. Hermann, R.: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. Am. Math. Soc. 11, 236–242 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270, A1645–A1648 (1970)

    MathSciNet  Google Scholar 

  18. Hirzebruch, F.: Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten. Math. Ann. 124, 77–86 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hwang, A.D., Simanca, S.R.: Distinguished Kähler metrics on Hirzebruch surfaces. Trans. Am. Math. Soc. 347(3), 1013–1021 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Hwang, A.D., Simanca, S.R.: Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics. Math. Ann. 309(1), 97–106 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ince, E.L.: Ordinary Differential Equations. Dover Publications, New York (1944)

    MATH  Google Scholar 

  22. Jelonek, W.: Extremal Kähler \(AC^{\perp }\)-surfaces. Bull. Belg. Math. Soc. Simon Stevin 9(4), 561–571 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Kamada, H.: Indefinite Kähler metrics of constant scalar curvature on Hirzebruch surfaces. Bull. Lond. Math. Soc. 41(6), 1060–1072 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kazdan, J.L., Warner, F.W.: Curvature functions for open 2-manifolds. Ann. Math. (2) 99, 203–219 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kobayashi, O.: Yamabe metrics and conformal transformations. Tôhoku Math. J. (2) 44(2), 251–258 (1992)

    Article  MATH  Google Scholar 

  26. Kobayashi, S.: Principal fibre bundles with the 1-dimensional toroidal group. Tôhoku Math. J. (2) 8, 29–45 (1956)

    Article  MATH  Google Scholar 

  27. Koiso, N., Sakane, Y.: Nonhomogeneous Kähler-Einstein Metrics on Compact Complex Manifolds. Lecture Notes in Mathematics, pp. 165–179. Springer, Berlin (1986)

    Google Scholar 

  28. Koiso, N., Sakane, Y.: Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II. Osaka J. Math. 25(4), 933–959 (1988)

    MathSciNet  MATH  Google Scholar 

  29. Lamari, A.: Courants kählériens et surfaces compactes. Ann. Inst. Fourier (Grenoble) 49(1), vii, x, 263–285 (1999)

    Google Scholar 

  30. LeBrun, C.: Counter-examples to the generalized positive action conjecture. Comm. Math. Phys. 118(4), 591–596 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. de Lima, L.L., Piccione, P., Zedda, M.: A note on the uniqueness of solutions for the Yamabe problem. Proc. Am. Math. Soc. 140(12), 4351–4357 (2012)

    Article  MATH  Google Scholar 

  33. Madsen, A.B., Pedersen, H., Poon, Y.S., Swann, A.: Compact Einstein-Weyl manifolds with large symmetry group. Duke Math. J. 88(3), 407–434 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Madsen, A.B., Pedersen, H., Poon, Y.S., Swann, A.: Corrigendum: compact Einstein-Weyl manifolds with large symmetry group. Duke Math. J. 100(1), 167 (1999)

    Article  MathSciNet  Google Scholar 

  35. Marrero, J.C., Rocha, J.: Locally conformal Kähler submersions. Geom. Dedicata 52(3), 271–289 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nakagawa, Y.: On generalized Kähler-Ricci solitons. Osaka J. Math. 48(2), 497–513 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Otoba, N.: New Examples of Riemannian Metrics with Constant Scalar Curvature. Master’s thesis, Keio University (2012)

  38. Page, D.: A compact rotating gravitational instanton. Phys. Lett. B 79(3), 235–238 (1978)

    Article  Google Scholar 

  39. Pedersen, H., Swann, A.: Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature. J. Reine Angew. Math. 441, 99–113 (1993)

    MathSciNet  MATH  Google Scholar 

  40. Petersen, P.: Riemannian Geometry, Grad. Texts in Math., 2nd edn. Springer, New York (2006)

    Google Scholar 

  41. Polyanin, A., Zaitsev, V.: Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton (2003)

    MATH  Google Scholar 

  42. Ritoré, M.: Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces. Comm. Anal. Geom. 9(5), 1093–1138 (2001)

    MathSciNet  MATH  Google Scholar 

  43. Sakane, Y.: Examples of compact Einstein Kähler manifolds with positive Ricci tensor. Osaka J. Math. 23(3), 585–616 (1986)

    MathSciNet  MATH  Google Scholar 

  44. Steenrod, N.: The topology of fibre bundles (Reprint of the 1957 edition), Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1999)

    Google Scholar 

  45. Vaisman, I.: On locally conformal almost Kähler manifolds. Israel J. Math. 24(3–4), 338–351 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vilms, J.: Totally geodesic maps. J. Differ. Geom. 4, 73–79 (1970)

    MathSciNet  MATH  Google Scholar 

  47. Watson, B.: Almost Hermitian submersions. J. Differ. Geom. 11(1), 147–165 (1976)

    MATH  Google Scholar 

  48. Whittaker, E.T., Watson, G.N.: A course of modern analysis (fourth edition, reprinted). Cambridge University Press, New York (1962)

    Google Scholar 

  49. Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Supér. (4) 8(4), 487–507 (1975)

    MATH  Google Scholar 

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Acknowledgments

A part of this paper is based on [37], written under the supervision of Professor H. Izeki. I express my gratitude to Professor O. Kobayashi for his genuine advice and encouragement after reading an earlier version of the thesis. Professor J. Viaclovsky kindly recommended [4] and [14], thereby reminding me that the constant scalar curvature metrics constructed in the thesis on \(\mathbb {CP}^2\#\overline{\mathbb {CP}}^2\) could be generalized to higher Hirzebruch surfaces, and that the corresponding \(B^t\)-flat analogue should be in presence. I am also grateful to Professor S. Shimomura for helping me understand the Bach-flat equation, and to Professor S. Matsuo for his advice on a preprint of this paper. This work was supported by Japan Society for the Promotion of Science under Research Fellowship for Young Scientists.

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  1. Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa , 223-8522, Japan

    Nobuhiko Otoba

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Otoba, N. Constant scalar curvature metrics on Hirzebruch surfaces. Ann Glob Anal Geom 46, 197–223 (2014). https://doi.org/10.1007/s10455-014-9419-z

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