Abstract
In this note, we look at estimates for the scalar curvature κof a compact, connected Riemannian manifold Mwhich are related to spinc Dirac operators.We show that one may not enlarge a Kähler metric with positiveRicci curvature without making κ smaller somewhere on M.More generally, if f: N → M is an area-nonincreasing map of a certain topological type,then the scalar curvature k of Ncannot be everywhere larger than κ ∘ f.If k ≥ κ ∘ f, then N is isometric to M × F, where F possesses a parallel untwisted spinor.
We also give explicit upper bounds for min κfor arbitrary Riemannian metrics on certainsubmanifolds of complex projective space.In certain cases, these estimates are sharp:we give examples where equality is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bär, C. and Bleecker, D.: The Dirac operator and the scalar curvature of continuously deformed algebraic varieties, in: Booss-Bavnbek, Bernhelm et al. (eds), Geometric Aspects of Partial Differential Equations, Contemp. Math. 242, Amer. Math. Soc., Providence, RI, 1998, pp. 3-24.
Besse, A. L.: Einstein Manifolds, Spinger-Verlag, Berlin, 1987.
Goette, S. and Semmelmann, U.: Scalar curvature estimates for compact symmetric spaces, Differential Geom. Appl., to appear.
Gromov, M.: Positive curvature, macroscopic dimension, spectral gaps and higher signatures, in: S. Gindikin, J. Lepowski and R. L. Wilson (eds), Functional Analysis on the Eve of the 21st Century, Vol. II, Progr. Math. 132, Birkhäuser, Boston, 1996, pp. 1-213.
Gromov, M. and Lawson Jr., H. B.: The classification of simply-connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423-434.
Hano, J.: Einstein complete intersections in complex projective spaces, Math. Ann. 216 (1975), 197-208.
Hirzebruch, F.: Topological Methods in Algebraic Geometry, 3rd enlarged edn, Springer, New York, 1966.
Hitchin, N.: Harmonic spinors, Adv. Math. 14 (1974), 1-55.
Kazdan, J. L. and Warner, F. W.: Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom. 10 (1975), 113-134.
Kobayashi, S. and Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31-47.
Kramer, W.: Der Dirac-Operator auf Faserungen, Dissertation, Bonner Mathematische Schriften 317, Universität Bonn, 1999.
Lawson Jr., H. B. and Michelsohn, M.-L.: Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989.
Llarull, M.: Scalar curvature estimates for (n+4k)-dimensional manifolds, Differential Geom. Appl. 6 (1996), 321-326.
Llarull, M.: Sharp estimates and the Dirac operator, Math. Ann. 310 (1998), 55-71.
Min-Oo, M.: Scalar curvature rigidity of certain symmetric spaces, in: F. Lalonde (ed.), Geometry, Topology and Dynamics, Montreal, PQ, 1995, CRM Proc. Lecture Notes 15, Amer. Math. Soc., Providence, RI, 1998, pp. 127-136.
Moroianu, A.: Parallel and Killing spinors on Spinc manifolds, Comm. Math. Phys. 187 (1997), 417-427.
Ogiue, K.: Scalar curvature of complex submanifolds in complex projective spaces, J. Differential Geom. 5 (1971), 229-232.
Schrödinger, E.: Diracsches Elektron im Schwerefeld. I, Sitzungsber. Preuss. Akad. Wiss. 11 (1932), 105-128.
Stolz, S.: Simply connected manifolds of positive scalar curvature, Bull. Amer. Math. Soc. 23 (1990), 427-432.
Stolz, S.: Positive scalar curvature metrics-Existence and classification questions, in: Proc. of the ICM, Vol. I, ICM Zürich, 1994, S. D. Chatterji (ed.), Birkhäuser, Basel, 1995, pp. 625-636.
Tian, G.: On Kähler-Einstein metrics on certain manifolds with c 1 (M) > 0, Invent. Math. 89 (1987), 225-246.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goette, S., Semmelmann, U. Spinc Structures and Scalar Curvature Estimates. Annals of Global Analysis and Geometry 20, 301–324 (2001). https://doi.org/10.1023/A:1013035721335
Issue Date:
DOI: https://doi.org/10.1023/A:1013035721335