Skip to main content
SpringerLink
Log in

Positive scalar curvature and the Dirac operator on complete riemannian manifolds

  • Published:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. S. Agmon,Elliptic Boundary Value Problems, Van Nostrand, 1965.

  2. F. J. Almgren, Jr., Existence and regularity almost everywhere to solutions of elliptic variational problems among surfaces of varying topological type and singularity structure,Ann. of Math. 87 (1968), 321–391.

    Article  MathSciNet  Google Scholar 

  3. M. F. Atiyah, Elliptic operators, discrete groups and Von Neumann algebras,Astérisque, vol. 32–33, Société Math. de France, 1976, 43–72.

    MathSciNet  Google Scholar 

  4. M. F. Atiyah, R. Bott andA. Shapiro, Clifford modules,Topology 3 (Suppl. 1) (1964), 3–38.

    Article  MathSciNet  Google Scholar 

  5. M. F. Atiyah andF. Hirzebruch, Vector bundles and homogeneous spaces,Proc. Symp. Pure Math. A.M.S. III (1961), 7–38.

  6. M. F. Atiyah andI. Singer, The index of elliptic operators I,Ann. of Math. 87 (1968), 484–530.

    Article  MathSciNet  Google Scholar 

  7. M. F. Atiyah andI. Singer, The index of elliptic operators III,Ann. of Math. 87 (1968), 546–604.

    Article  MathSciNet  Google Scholar 

  8. M. F. Atiyah andI. Singer, The index of elliptic operators V,Ann. of Math. 93 (1971), 139–149.

    Article  MathSciNet  Google Scholar 

  9. M. F. Atiyah andI. Singer, Index Theory for skew-adjoint Fredholm operators,Publ. Math. I.H.E.S. 37, (1969), 1–26.

    Google Scholar 

  10. L. Bérard Bergery, La courbure scalaire des variétés riemanniennes, Sém. Bourbaki, exposé no 556, juin 1980,Springer Lecture Notes 842 (1981), 225–245.

    Google Scholar 

  11. L. Bérard Bergery,Scalar curvature and isometry group (to appear).

  12. Yu. D. Burago andV. A. Toponagov, On 3-dimensional riemannian spaces with curvature bounded above,Math. Zametki 13 (1973), 881–887.

    MATH  Google Scholar 

  13. Yu. D.Burago andV. A. Zagallar,Geometric Inequalities (in Russian), Leningrad, 1980.

  14. I. Chavel andE. Feldman,Isoperimetric inequalities on curved surfaces (to appear).

  15. J. Cheeger andD. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature,J. Diff. Geom. 6 (1917), 119–128.

    MathSciNet  Google Scholar 

  16. S. Y. Cheng andS. T. Yau, Differential equations on riemannian manifolds and their geometric applications,Comm. Pure and Appl. Math. 28 (1975), 333–354.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Federer,Geometric Measure Theory, N.Y., Springer Verlag, 1969.

    MATH  Google Scholar 

  18. H. Federer, Real flat chains, cochains and variational problems,Indiana Univ. Math. J. 24 (1974), 351–407.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Federer andW. Fleming, Normal and integral currents,Ann. of Math. 72 (1960), 458–520.

    Article  MathSciNet  Google Scholar 

  20. F. Fiala, Le problème des isopérimètres sur les surfaces ouvertes à courbure positive,Comm. Math. Helv. 13 (1941), 293–346.

    Article  MathSciNet  Google Scholar 

  21. D. Fischer-Colbrie andR. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature,Comm. Pure and Appl. Math. 33 (1980), 199–211.

    Article  MATH  MathSciNet  Google Scholar 

  22. W. Fleming, On the oriented Plateau problem,Rend. Circ. Mat. Palermo (2)11 (1962), 69–90.

    Article  MATH  MathSciNet  Google Scholar 

  23. Z. Gao,Thesis, Stony Brook, 1982.

  24. M. Gromov, Filling riemannian manifolds,J. Diff. Geom. 18 (1983), 1–147.

    MATH  MathSciNet  Google Scholar 

  25. M. Gromov andH. B. Lawson, Jr., Spin and scalar curvature in the presence of a fundamental group,Ann. of Math. 111 (1980), 209–230.

    Article  MathSciNet  Google Scholar 

  26. M. Gromov andH. B. Lawson, Jr., The classification of simply-connected manifolds of positive scalar curvature,Ann. of Math. 111 (1980), 423–434.

    Article  MathSciNet  Google Scholar 

  27. R. Hardt andL. Simon, Boundary regularity and embedded solutions for the oriented Plateau problem,Ann. of Math. 110 (1979), 439–486.

    Article  MathSciNet  Google Scholar 

  28. P. Hartman, Geodesic parallel coordinates in the large,Amer. J. Math. 86 (1964), 705–727.

    Article  MathSciNet  Google Scholar 

  29. N. Hitchin, Harmonic spinors,Adv. in Math. 14 (1974), 1–55.

    Article  MATH  MathSciNet  Google Scholar 

  30. V. K. Ionin, Isoperimetric and certain other inequalities for a manifold of bounded curvature,Sibirski Math. J. 10 (1969), 329–342.

    MATH  MathSciNet  Google Scholar 

  31. J. Kazdan, Deforming to positive scalar curvature on complete manifolds,Math. Ann. 261 (1982), 227–234.

    Article  MATH  MathSciNet  Google Scholar 

  32. J. Kazdan andF. Warner, Prescribing curvatures,Proc. of Symp. in Pure Math. 27 (1975), 309–319.

    MathSciNet  Google Scholar 

  33. S. Kobayashi andK. Nomizu,Foundations of Differential Geometry, N.Y., Interscience, 1963.

    MATH  Google Scholar 

  34. H. B. Lawson Jr., Minimal Varieties,Proc. of Symp. in Pure Math. 27 (1975), 143–175.

    MathSciNet  Google Scholar 

  35. H. B. Lawson Jr., The unknottedness of minimal embeddings,Inv. Math. 11 (1970), 183–187.

    Article  MATH  MathSciNet  Google Scholar 

  36. H. B. Lawson Jr. andM. L. Michelsohn,Spin Geometry (to appear).

  37. H. B. Lawson Jr. andS. T. Yau, Scalar curvature, non-abelian group actions and the degree of symmetry of exotic spheres,Comm. Math. Helv. 49 (1974), 232–244.

    Article  MATH  MathSciNet  Google Scholar 

  38. A. Lichnerowicz, Spineurs harmoniques,C.r. Acad. Sci. Paris, Sér. A-B,257 (1963), 7–9.

    MATH  MathSciNet  Google Scholar 

  39. G. Lusztig, Novikov’s higher signature and families of elliptic operators,J. Diff. Geom. 7 (1971), 229–256.

    MathSciNet  Google Scholar 

  40. W. Meeks III,L. Simon andS. T. Yau, Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature,Ann. of Math. 116 (1982), 621–659.

    Article  MathSciNet  Google Scholar 

  41. J. Milnor, On manifolds homemorphic to the 7-sphere,Ann. of Math. (2)64 (1956), 399–405.

    Article  MathSciNet  Google Scholar 

  42. J. Milnor, Remarks concerning spin manifolds, “Differential Geometry and Combinatorial Topology”, Princeton University Press, 1965, 55–62.

  43. J. Milnor,Topology from the Differentiable Viewpoint, Univ. of Virginia Press, Charlottesville, 1965.

    MATH  Google Scholar 

  44. J. Milnor, A Unique factorization theorem for 3-manifolds,Amer. J. Math. 84 (1962), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  45. T. Miyazaki,On the existence of positive scalar curvature metrics on non-simply-connected manifolds (to appear).

  46. C. B. Morrey,Multiple Integrals and the Calculus of Variations, New York, Springer-Verlag, 1966.

    Google Scholar 

  47. R. Osserman, Bonneson-style isoperimetric inequalities,Amer. Math. Monthly 86 (1979), 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  48. T. Sakai,On a theorem of Burago-Toponogov (to appear).

  49. H. H. Schaefer,Topological Vector Spaces, N.Y., Springer-Verlag, 1971.

    Google Scholar 

  50. R. Schoen,A remark on minimal hypercones (to appear).

  51. R. Schoen andS. T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature,Ann. of Math. 110 (1979), 127–142.

    Article  MathSciNet  Google Scholar 

  52. R. Schoen andS. T. Yau, On the structure of manifolds with positive scalar curvature,Manuscripta Math. 28 (1979), 159–183.

    Article  MATH  MathSciNet  Google Scholar 

  53. R. Schoen andS. T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature,in Seminar on differential geometry,Annals of Math. Studies, no 102, Princeton Univ. Press, 1982, 209–227.

  54. R. Schoen andS. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature,Comm. Math. Helv. 39 (1976), 333–341.

    Article  MathSciNet  Google Scholar 

  55. R. Schoen andS. T. Yau (to appear).

  56. J. Simons, Minimal varieties in riemannian manifolds,Ann. of Math. 88 (1968), 62–105.

    Article  MathSciNet  Google Scholar 

  57. R. Thom, Quelques propriétés globales des variétés différentiables,Comm. Math. Helv. 28 (1954), 17–86.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut des Hautes Études Scientifiques, 35, route de Chartres, 91440, Bures-sur-Yvette

    Mikhael Gromov & H. Blaine Lawson

  2. Department of Mathematics, State University of New York at Stony Brook, 11794, Stony Brook, N.Y.

    Mikhael Gromov & H. Blaine Lawson

Authors
  1. Mikhael Gromov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Blaine Lawson
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by NSF Grant number MCS 830 1365.

About this article

Cite this article

Gromov, M., Lawson, H.B. Positive scalar curvature and the Dirac operator on complete riemannian manifolds. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 58, 83–196 (1983). https://doi.org/10.1007/BF02953774

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02953774

Keywords

Search

Navigation