Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T18:33:01.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal Sasakian geometry on ${T}^{2} \times {S}^{3} $ and related manifolds

Part of: Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  03 June 2013

Charles P. Boyer  and
Christina W. Tønnesen-Friedman
Charles P. Boyer
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA email cboyer@math.unm.edu
Christina W. Tønnesen-Friedman
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308, USA email tonnesec@union.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on ${T}^{2} \times {S}^{3} $ and certain related 5-manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.

Keywords

extremal Sasaki metricsextremal Kähler metricsSasaki cone and Sasaki bouquetjoin construction

MSC classification

Secondary: 53D42: Symplectic field theory; contact homology 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1431 - 1456
Copyright
© The Author(s) 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abreu, M., Kähler metrics on toric orbifolds, J. Differential Geom. 58 (2001), 151–187; MR 1895351 (2003b:53046).Google Scholar
Ahara, K. and Hattori, A., 4-dimensional symplectic ${S}^{1} $-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), 251–298; MR 1127083 (93b:58048).Google Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C. W., Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differential Geom. 68 (2004), 277–345; MR 2144249.CrossRefGoogle Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C. W., Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability, Invent. Math. 173 (2008), 547–601; MR 2425136 (2009m:32043).Google Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C. W., Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds, Comm. Anal. Geom. 16 (2008), 91–126; MR 2411469 (2010c:32043).Google Scholar
Apostolov, V., Calderbank, D. M. J., Gauduchon, P. and Tønnesen-Friedman, C. W., Extremal Kähler metrics on projective bundles over a curve, Adv. Math. 227 (2011), 2385–2424.CrossRefGoogle Scholar
Atiyah, M. F., Complex fibre bundles and ruled surfaces, Proc. Lond. Math. Soc. (3) 5 (1955), 407–434; MR 0076409 (17,894b).Google Scholar
Atiyah, M. F., Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3) 7 (1957), 414–452; MR 0131423 (24 #A1274).CrossRefGoogle Scholar
Audin, M., Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension 4, in Géométrie symplectique et mécanique, La Grande Motte, 1988, Lecture Notes in Mathematics, vol. 1416 (Springer, Berlin, 1990), 1–25; MR 1047474 (91f:57013).Google Scholar
Audin, M., Torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, revised edition (Birkhäuser, Basel, 2004); MR 2091310 (2005k:53158).Google Scholar
Boyer, C. P., The Sasakian geometry of the Heisenberg group, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52 (2009), 251–262; MR 2554644.Google Scholar
Boyer, C. P., Completely integrable contact Hamiltonian systems and toric contact structures on ${S}^{2} \times {S}^{3} $, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 058, 22.Google Scholar
Boyer, C. P., Extremal Sasakian metrics on ${S}^{3} $-bundles over ${S}^{2} $, Math. Res. Lett. 18 (2011), 181–189; MR 2756009 (2012d:53132).CrossRefGoogle Scholar
Boyer, C. P., Maximal tori in contactomorphism groups, Differential Geom. Appl. 31 (2013), 190–216.CrossRefGoogle Scholar
Boyer, C. P. and Galicki, K., On Sasakian-Einstein geometry, Internat. J. Math. 11 (2000), 873–909; MR 2001k:53081.Google Scholar
Boyer, C. P. and Galicki, K., A note on toric contact geometry, J. Geom. Phys. 35 (2000), 288–298; MR 1780757 (2001h:53124).Google Scholar
Boyer, C. P. and Galicki, K., Sasakian geometry, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2008); MR 2382957 (2009c:53058).Google Scholar
Boyer, C. P., Galicki, K. and Ornea, L., Constructions in Sasakian geometry, Math. Z. 257 (2007), 907–924; MR 2342558 (2008m:53103).CrossRefGoogle Scholar
Boyer, C. P., Galicki, K. and Simanca, S. R., Canonical Sasakian metrics, Comm. Math. Phys. 279 (2008), 705–733; MR 2386725.Google Scholar
Boyer, C. P. and Pati, J., On the equivalence problem for toric contact structures on ${S}^{3} $-bundles over ${S}^{2} $, Preprint (2012), arXiv:math.SG/1204.2209.Google Scholar
Boyer, C. P. and Tønnesen-Friedman, C. W., Extremal Sasakian geometry on ${S}^{3} $-bundles over Riemann surfaces, Preprint (2013), arXiv:1302.0776.Google Scholar
Bryant, R. L., Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), 623–715; MR 1824987 (2002i:53096).Google Scholar
Burns, D. and De Bartolomeis, P., Stability of vector bundles and extremal metrics, Invent. Math. 92 (1988), 403–407; MR 936089 (89d:53114).Google Scholar
Buşe, O., Deformations of Whitehead products, symplectomorphism groups, and Gromov-Witten invariants, Int. Math. Res. Not. IMRN (2010), 3303–3340; MR 2680275.Google Scholar
Calabi, E., Extremal Kähler metrics, in Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102 (Princeton University Press, Princeton, NJ, 1982), 259–290; MR 83i:53088.Google Scholar
Calabi, E., Extremal Kähler metrics. II, in Differential geometry and complex analysis (Springer, Berlin, 1985), 95–114; MR 780039 (86h:53067).CrossRefGoogle Scholar
David, L. and Gauduchon, P., The Bochner-flat geometry of weighted projective spaces, in Perspectives in Riemannian geometry, CRM Proceedings & Lecture Notes, vol. 40 (American Mathematical Society, Providence, RI, 2006), 109–156; MR 2237108 (2007h:32032).Google Scholar
Folland, G. B., Compact Heisenberg manifolds as CR manifolds, J. Geom. Anal. 14 (2004), 521–532; MR 2077163 (2005d:32057).Google Scholar
Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 105–165; MR 90d:53083.Google Scholar
Fujiki, A., Remarks on extremal Kähler metrics on ruled manifolds, Nagoya Math. J. 126 (1992), 89–101; MR 1171594 (94c:58033).Google Scholar
Gauduchon, P., Calabi’s extremal Kähler metrics, preliminary version, (2010).Google Scholar
Ghigi, A. and Kollár, J., Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007), 877–902; MR 2341843 (2008j:32027).Google Scholar
Gorbacevič, V. V., Compact homogeneous spaces of dimension five and higher, Uspehi Mat. Nauk 33 (1978), 161–162; MR 0500718 (58 #18279).Google Scholar
Hwang, A. D., On existence of Kähler metrics with constant scalar curvature, Osaka J. Math. 31 (1994), 561–595; MR 1309403 (96a:53061).Google Scholar
Karshon, Y., Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999); MR 1612833 (2000c:53113).Google Scholar
Kodaira, K., Nirenberg, L. and Spencer, D. C., On the existence of deformations of complex analytic structures, Ann. of Math. (2) 68 (1958), 450–459; MR 0112157 (22 #3012).CrossRefGoogle Scholar
Kodaira, K. and Spencer, D. C., A theorem of completeness for complex analytic fibre spaces, Acta Math. 100 (1958), 281–294; MR 0112155 (22 #3010).Google Scholar
Kreck, M. and Lück, W., Topological rigidity for non-aspherical manifolds, Pure Appl. Math. Q. 5 (2009), 873–914; Special issue: in honor of Friedrich Hirzebruch. Part 2; MR 2532709 (2010g:57026).Google Scholar
Lee, J. M., CR manifolds with noncompact connected automorphism groups, J. Geom. Anal. 6 (1996), 79–90; MR 1402387 (97h:32013).CrossRefGoogle Scholar
Lerman, E., Contact toric manifolds, J. Symplectic Geom. 1 (2002), 785–828; MR 2 039 164.Google Scholar
Lerman, E., On maximal tori in the contactomorphism groups of regular contact manifolds, Preprint (2002), arXiv:math.SG/0212043.Google Scholar
Lerman, E., Homotopy groups of $K$-contact toric manifolds, Trans. Amer. Math. Soc. 356 (2004), 4075–4083; MR 2 058 839.Google Scholar
Lutz, R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble) 29 (1979), 283–306; MR 82j:53067.Google Scholar
Maruyama, M., On automorphism groups of ruled surfaces, J. Math. Kyoto Univ. 11 (1971), 89–112; MR 0280493 (43 #6213).Google Scholar
McDuff, D., Notes on ruled symplectic 4-manifolds, Trans. Amer. Math. Soc. 345 (1994), 623–639; MR 1188638 (95a:57034).Google Scholar
McDuff, D., Symplectomorphism groups and almost complex structures, in Essays on geometry and related topics, Vols. 1, 2, Monographies de L’Enseignement Mathématique, vol. 38 (Enseignement Math, Geneva, 2001), 527–556; MR 1929338 (2003i:57042).Google Scholar
McDuff, D. and Salamon, D., Introduction to symplectic topology, Oxford Mathematical Monographs, second edition (The Clarendon Press Oxford University Press, New York, 1998); MR 2000g:53098.Google Scholar
Morrow, J. and Kodaira, K., Complex manifolds (AMS Chelsea Publishing, Providence, RI, 2006), Reprint of the 1971 edition with errata; MR 2214741 (2006j:32001).Google Scholar
Schoen, R., On the conformal and CR automorphism groups, Geom. Funct. Anal. 5 (1995), 464–481; MR 1334876 (96h:53047).Google Scholar
Suwa, T., On ruled surfaces of genus 1, J. Math. Soc. Japan 21 (1969), 291–311; MR 0242198 (39 #3531).CrossRefGoogle Scholar
Tønnesen-Friedman, C. W., Extremal Kähler metrics on minimal ruled surfaces, J. Reine Angew. Math. 502 (1998), 175–197; MR 1647571 (99g:58026).Google Scholar
Tønnesen-Friedman, C. W., Extremal Kähler metrics and Hamiltonian functions. II, Glasg. Math. J. 44 (2002), 241–253; MR 1902401 (2003c:58011).CrossRefGoogle Scholar