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The Sasaki Cone and Extremal Sasakian Metrics

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Book cover Riemannian Topology and Geometric Structures on Manifolds

Part of the book series: Progress in Mathematics ((PM,volume 271))

Abstract

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone and use its critical points to single out the strongly extremal Reeb vector fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest L 2-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five-dimensional case.

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References

  1. A. Bangaya & P. Molino, G éométric des formes de contact complètement intégrables de type toriques, Séminaire G. Darboux de Géométrie et Topologie Différentielle, 1991–1992, Montpellier.

    Google Scholar 

  2. D. Barden, Simply connected five-manifolds, Ann. Math. (2) 82 (1965), pp. 365–385. MR 32 #1714.

    Article  MathSciNet  Google Scholar 

  3. C.P. Boyer & K. Galicki, A note on toric contact geometry, J. Geom. Phys. 35 (2000), 4, pp. 288–298. MR 2001h:53124.

    Article  MATH  MathSciNet  Google Scholar 

  4. ——, Einstein metrics on rational homology spheres, J. Diff. Geom. (3) 74 (2006), pp. 353– 362. MR MR2269781.

    MATH  MathSciNet  Google Scholar 

  5. ——, New Einstein metrics in dimension five, J. Diff. Geom. (3) 57 (2001), pp. 443–463. MR 2003b:53047.

    MATH  MathSciNet  Google Scholar 

  6. ——, On Sasakian-Einstein geometry, Internat. J. Math. (7) 11 (2000), pp. 873–909. MR 2001k:53081.

    Article  MATH  MathSciNet  Google Scholar 

  7. ——, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, 2008, MR2382957.

    Google Scholar 

  8. ——, Sasakian geometry, hypersurface singularities, and Einstein metrics, Rend. Circ. Mat. Palermo (2) Suppl. (2005), 75, suppl., pp. 57–87. MR 2152356.

    Google Scholar 

  9. C.P. Boyer, K. Galicki & J. Kollár, Einstein metrics on spheres, Ann. Math. (2) 162 (2005), no. 1, pp. 557–580. MR 2178969(2006j:53058).

    Article  MATH  Google Scholar 

  10. C.P. Boyer, K. Galicki, J. Kollár & E. Thomas, Einstein metrics on exotic spheres in dimensions 7, 11, and 15, Experiment. Math. 14 (2005), no. 1, pp. 59–64. MR 2146519 (2006a:53042).

    Google Scholar 

  11. C.P. Boyer, K. Galicki & P. Matzeu, On eta-Einstein Sasakian geometry, Commun. Math. Phys. (1) 262 (2006), pp. 177–208. MR 2200887.

    Article  MATH  MathSciNet  Google Scholar 

  12. C.P. Boyer, K. Galicki & L. Ornea, Constructions in Sasakian Geometry, Math. Zeit. (4) 257 (2007), pp. 907–924. MR2342558.

    Article  MATH  MathSciNet  Google Scholar 

  13. C.P. Boyer, K. Galicki & S.R. Simanca, Canonical Sasakian metrics, Commun. Math. Phys. (3) 279 (2008), pp. 705–733. MR2386725.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Calabi, Extremal Kähler metrics II, in Differential geometry and complex analysis (I. Chavel & H.M. Farkas eds.), Springer-Verlag, Berlin, 1985, pp. 95–114.

    Google Scholar 

  15. K. Cho, A. Futaki & H. Ono, Uniqueness and examples of compact toric Sasaki–Einstein metrics, Comm. Math. Phys. (2) 277 (2008), pp. 439–458. MR2358291 (2008j:53076).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math., 73 (1983), pp. 437–443. MR 84j:53072.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Futaki, H. Ono & G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds, preprint arXiv:math. DG/0607586, (unpublished).

    Google Scholar 

  18. J.P. Gauntlett, D. Martelli, J. Sparks & W. Waldram, Sasaki–Einstein metrics on S 2 ×S 3, Adv. Theor. Math. Phys., 8 (2004), pp. 711–734.

    MATH  MathSciNet  Google Scholar 

  19. J.P. Gauntlett, D. Martelli, J. Sparks & S.-T. Yau, Obstructions to the existence of Sasaki– Einstein metrics, Comm. Math. Phys. (3) 273 (2007), pp. 803–827.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Ghigi & J. Kollár, K ähler–Einstein metrics on orbifolds and Einstein metrics on Spheres, Comment. Math. Helvetici (4) 82 (2007), pp. 877–902. MR2341843 (2008j:32027).

    Article  MATH  Google Scholar 

  21. G. Grantcharov & L. Ornea, Reduction of Sasakian manifolds, J. Math. Phys. (8) 42 (2001), pp. 3809–3816. MR 1845220 (2002e:53060).

    Article  MATH  MathSciNet  Google Scholar 

  22. A.R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, pp. 101–173, Cambridge Univ. Press, Cambridge, (2000).

    Google Scholar 

  23. J.M. Johnson & J. Kollár, K ähler–Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces, Ann. Inst. Fourier (Grenoble) (1) 51 (2001), pp. 69–79. MR 2002b:32041.

    MATH  MathSciNet  Google Scholar 

  24. J. Kollár, Circle actions on simply connected 5-manifolds, Topology (3) 45 (2006), pp. 643– 671. MR 2218760.

    Article  MATH  MathSciNet  Google Scholar 

  25. ——, Einstein metrics on five-dimensional Seifert bundles, J. Geom. Anal. (3) 15 (2005), pp. 445–476. MR 2190241.

    MATH  MathSciNet  Google Scholar 

  26. ——, Positive Sasakian structures on 5-manifolds, these Proceedings, Eds. Galicki & Simanca, Birkhauser, Boston.

    Google Scholar 

  27. C. LeBrun & S.R. Simanca, On the Kähler Classes of Extremal Metrics, Geometry and Global Analysis (Sendai, Japan 1993), First Math. Soc. Japan Intern. Res. Inst. Eds. Kotake, Nishikawa & Schoen.

    Google Scholar 

  28. E. Lerman, Contact toric manifolds, J. Symp. Geom. (4) 1 (2002), pp. 785–828, MR 2 039 164.

    MathSciNet  Google Scholar 

  29. D. Martelli, J. Sparks & S.-T. Yau, Sasaki–Einstein manifolds and volume minimisation, Comm. Math. Phys. (3) 280 (2008), pp. 611–673. MR2399609.

    Article  MATH  MathSciNet  Google Scholar 

  30. J.W. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press, Princeton, N.J., 1968. MR 39 #969.

    MATH  Google Scholar 

  31. J. Milnor & P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), pp. 385–393. MR 45 #2757.

    Article  MATH  MathSciNet  Google Scholar 

  32. P. Orlik, On the homology of weighted homogeneous manifolds, Proceedings of the 2nd Conference on Compact Transformation Groups (Univ. of Mass., Amherst, Mass., 1971), Part I, Lect. Notes Math. 298, Springer-Verlag, Berlin, 1972, pp. 260–269. MR 55 #3312.

    Chapter  Google Scholar 

  33. ——, Seifert manifolds, Lect. Notes Math. 291, Springer-Verlag, Berlin, 1972, MR 54 #13950.

    MATH  Google Scholar 

  34. P. Orlik & R.C. Randell, The monodromy of weighted homogeneous singularities, Invent. Math. (3) 39 (1977), pp. 199–211. MR 57 #314.

    Article  MATH  MathSciNet  Google Scholar 

  35. P. Orlik & P. Wagreich, Isolated singularities of algebraic surfaces with C* action, Ann. Math. (2) 93 (1971), pp. 205–228. MR 44 #1662.

    Article  MathSciNet  Google Scholar 

  36. R.C. Randell, The homology of generalized Brieskorn manifolds, Topology (4) 14 (1975), pp. 347–355. MR 54 #1270.

    Article  MATH  MathSciNet  Google Scholar 

  37. P. Rukimbira, Chern-Hamilton’s conjecture and K-contactness, Houston J. Math. (4) 21 (1995), pp. 709–718. MR 96m:53032.

    MATH  MathSciNet  Google Scholar 

  38. S.R. Simanca, Canonical metrics on compact almost complex manifolds, Publicacões Matemáticas do IMPA, IMPA, Rio de Janeiro (2004), 97 pp.

    Google Scholar 

  39. ——, Heat Flows for Extremal Kähler Metrics, Ann. Scuola Norm. Sup. Pisa CL. Sci., 4 (2005), pp. 187–217.

    MATH  MathSciNet  Google Scholar 

  40. ——, Precompactness of the Calabi Energy, Internat. J. Math., 7 (1996) pp. 245–254.

    Article  MATH  MathSciNet  Google Scholar 

  41. ——, Strongly Extremal Kähler Metrics, Ann. Global Anal. Geom. 18 (2000), no. 1, pp. 29–46.

    Article  MATH  MathSciNet  Google Scholar 

  42. S.R. Simanca & L.D. Stelling, Canonical Kähler classes. Asian J. Math. 5 (2001), no. 4, pp. 585–598.

    MATH  MathSciNet  Google Scholar 

  43. S. Smale, On the structure of 5-manifolds, Ann. Math. (2) 75 (1962), pp. 38–46. MR 25 #4544.

    Article  MathSciNet  Google Scholar 

  44. T. Takahashi, Deformations of Sasakian structures and its application to the Brieskorn manifolds, Tôhoku Math. J. (2) 30 (1978), no. 1, pp. 37–43. MR 81e:53024.

    Article  Google Scholar 

  45. M.Y. Wang & W. Ziller, Einstein metrics on principal torus bundles, J. Diff. Geom. (1) 31 (1990), pp. 215–248. MR 91f:53041.

    MATH  MathSciNet  Google Scholar 

  46. S.S.-T. Yau and Y. Yu, Classification of 3-dimensional isolated rational hypersurface singularities with C*-action, Rocky Mountain J. Math. (5) 35 (2005), pp. 1795–1809. MR 2206037(2006j:32034).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    Charles P. Boyer, Krzysztof Galicki & Santiago R. Simanca

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  1. Charles P. Boyer
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  2. Krzysztof Galicki
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  3. Santiago R. Simanca
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  1. Department of Mathematics, University of New Mexico, Albuquerque, NM, 87131

    Krzysztof Galicki  & Santiago R. Simanca  & 

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Boyer, C.P., Galicki, K., Simanca, S.R. (2009). The Sasaki Cone and Extremal Sasakian Metrics. In: Galicki, K., Simanca, S.R. (eds) Riemannian Topology and Geometric Structures on Manifolds. Progress in Mathematics, vol 271. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4743-8_11

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