PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2014 | 1 | 1 |
Tytuł artykułu

On the transverse Scalar Curvature of a Compact Sasaki Manifold

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group
Wydawca
Czasopismo
Rocznik
Tom
1
Numer
1
Opis fizyczny
Daty
otrzymano
2013-10-14
zaakceptowano
2014-04-18
online
2014-09-19
Twórcy
autor
  • Department of Mathematics, University of Oregon, Eugene, OR, 97403,, whe@uoregon.edu
Bibliografia
  • [1] A.L. Besse, Einstein manifolds, Springer, 2nd edition.
  • [2] C.P. Boyer, K. Galicki; Sasaki geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008. xii+613 pp.
  • [3] C.P. Boyer, K. Galicki, S.R. Simanca; Canonical Sasaki metrics, Comm. Math. Phys. 279 (2008), no. 3, 705-733.
  • [4] C.P. Boyer, K. Galicki, S.R. Simanca; The Sasaki cone and extremal Sasaki metrics, Riemannian topology and geometric structures on manifolds, 263-290, Progr. Math., 271, Birkh äuser Boston, Boston, MA, 2009.
  • [5] E. Calabi; Extremal Kähler metric, in Seminar of Differential Geometry, ed. S. T. Yau, Annals of Mathematics Studies 102, Princeton University Press (1982), 259-290.
  • [6] E. Cakabi; Extremal Kähler metrics II, Differential geometry and complex analysis, 95-114, Springer, Berlin, 1985.
  • [7] E. Calabi, X. Chen; The space of Kähler metrics II, J. Differential Geom. 61 (2002), no. 2, 173-193.
  • [8] X. Chen; The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189-234.
  • [9] T. Collins, G. Szekelyhidi, K-Semistability for irregular Sasakian manifolds, arxiv.org/abs/1204.2230.
  • [10] S.K. Donaldson; Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists’ lectures, 384-403, World Sci. Ser. 20th Century Math., 5, World Sci. Publ., River Edge, NJ, 1997.
  • [11] S.K. Donaldson; Symmetric spaces, Kähler geometry and Hamiltonian dynamics. Northern California Symplectic Geometry Seminar, 13-33, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999.
  • [12] S.K. Donaldson; Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479-522.
  • [13] S.K. Donaldson; Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289-349.
  • [14] S.K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal. 19 (2009), 83-136.[Crossref]
  • [15] A. El Kacimi-Alaoui; Operateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 79, (1990), 57-106.
  • [16] A. Fujiki; The moduli spaces and Kähler metrics of polarised algebraic varieties, Suguku 42 (1990), 231-243; English transl., Sugaku Expositions 5 (1992), 173-191.
  • [17] A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), no. 3, 437-443.
  • [18] A. Futaki, H. Ono, G. Wang; Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom. 83 (2009), 585-636.
  • [19] J. P. Gauntlett, D. Martelli, J. Sparks, W. Waldram, Sasaki-Einstein metrics on S2 × S3, Adv. Theor. Math. Phys., 8 (2004), 711-734.
  • [20] J. P. Gauntlett, D. Martelli, J. Sparks, S.T. Yau; Obstructions to the Existence of Sasaki-Einstein Metrics, Commun. Math. Phy. 273 (2007), 803-827.
  • [21] P. Guan, X. Zhang; A geodesic equation in the space of Sasaki metrics, to appear in Yau’s Preceedings.
  • [22] P. Guan, X. Zhang; Regularity of the geodesic equation in the space of Sasaki metrics, arXiv:0906.5591.
  • [23] W. He, The Sasaki-Ricci ffow and compact Sasaki manifolds with positive transverse bisectional curvature, arXiv:1103.5807.
  • [24] E. Legendre, Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics, Compositio Mathematica 147 (2011), pp. 1613-1634[WoS]
  • [25] A. Lichnerowicz, Sur les transformations analytiques des variétés káhlériennes compactes, (French) C. R. Acad. Sci. Paris 244 1957 3011-3013.
  • [26] T. Mabuchi; K-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), no. 4, 575-593.
  • [27] T. Mabuchi; Some symplectic geometry on compact K¨hler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227-252.
  • [28] T. Mabuchi; Stability of extremal Kähler manifolds, Osaka J. Math. 41 (2004), no. 3, 563-582.
  • [29] D. Martelli, J. Sparks, S.T. Yau; Sasaki-Einstein Manifolds and Volume Minimisation, Commun.Math.Phys. 280 (2008), 611-673.
  • [30] Y. Matsushima, Sur la structure du groupe d’homéomorphismes analytiques dùne certaine variété kaehlérienne, Nagoya Math. J. 11 (1957), 145-150.
  • [31] D. McDuff, D. Salamon; Introduction to symplectic topology, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. viii+425 pp.
  • [32] J. Ross, R. Thomas; Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics, arXiv:0907.5214.
  • [33] S. Semmes; Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495-550.
  • [34] J. Sparks, Sasakian-Einstein manifolds, arXiv:1004.2461.
  • [35] G. Székelyhidi; Extremal metrics and K-stability. Bull. Lond. Math. Soc. 39 (2007), no. 1, 76-84.
  • [36] G. Tian; Kähler-Einstein metrics with positive scalar curvature Invent. Math. 130 (1997), no. 1, 1-37.
  • [37] S.T. Yau; Open problems in geometry. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), 1-28, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_coma-2014-0004
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.