Skip to main content
SpringerLink
Log in

Disconnectedness of sublevel sets of some Riemannian functionals

  • Published:
Geometric & Functional Analysis GAFA Aims and scope Submit manuscript

Abstract

LetM be a compact manifold of dimension greater than four. Denote byRiem(M) the space of Riemannian structures onM (i.e. of isometry classes of Riemannian metrics onM) endowed with the Gromov-Hausdorff metric. LetRiem ε(M) ⊂Riem(M) be its subset formed by all Riemannian structuresμ such that vol(μ)=1 andinj(μ) ≥ε, whereinj(μ) denotes the injectivity radius ofμ.

We prove that for all sufficiently small positiveε the spaceRiem ε(M) is disconnected. Moreover, ifε is sufficiently small, thenRiem ε(M) is representable as the union of two non-empty subsetsA andB such that the Gromov-Hausdorff distance between any element ofA and any element ofB is greater thanε/9. We also prove a more general result with the following informal meaning: There exist two Riemannian structures of volume one and arbitrarily small injectivity radius onM such that any continuous path (and even any sequence of sufficiently small “jumps”) in the space of Riemannian structures of volume one onM connecting these Riemannian structures must pass through Riemannian structures of injectivity radius “uncontrollably” smaller than the injectivity radii of these two Riemannian structures.

These results can be generalized for at least some four-dimensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Riemannian structures onM formed by imposing various constraints on curvatures, volume, diameter, etc.

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

Similar content being viewed by others

References

  1. F. Acquistapace, R. Benedetti, F. Broglia, Effectiveness-non-effectiveness in semi-algebraic and PL geometry, Inv. Math. 102:1 (1990), 141–156.

    Google Scholar 

  2. M. Anderson, J. Cheeger,C α-compactness for manifolds with Ricci curvature and injectivity radius bounded from below, J. Diff. Geom. 35 (1992), 265–281.

    Google Scholar 

  3. J. Bochnak, M. Coste, M.-F. Roy, Geometrie Algebrique Reelle, Springer, 1987.

  4. G. Boolos, R. Jeffrey, Computability and Logic”, Third Edition, Cambridge University Press, 1989.

  5. W. Boone, W. Haken, V. Poenaru, On recursively unsolvable problems in topology and their classification, in “Contributions to Mathematical Logic” (H. Arnold Schmidt, K. Schutte, H.-J. Thiele, eds.), North-Holland, 1968.

  6. Yu. Burago, M. Gromov, G. Perelman, A.D. Alexandrov spaces with curvature bounded below, Russ. Math. Surv. 47 (1992), 1–58.

    Google Scholar 

  7. J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74.

    Google Scholar 

  8. J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded I, II”, J. Diff. Geom. 23 (1986), 309–346 and J. Diff. Geom. 32 (1990), 269–298.

    Google Scholar 

  9. M. Coste, Ensembles semi-algebriques, in “Geometrie algebrique reelle et formes quadratiques”, Journees S.M.F. Universite de Rennes, (J.-L. Colliot-Thelene, M. Coste, L. Mahe, M.-F. Roy, eds.) Springer Lecture Notes in Math. 959 (1982), 109–138.

    Google Scholar 

  10. C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Scient. Ec. Norm. Sup. 13 (1980), 419–435.

    Google Scholar 

  11. H.B. Enderton, Elements of recursion theory, in “Handbook of Mathematical Logic”, (J. Barwise, ed.), North-Holland, 1977, 527–566.

  12. K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, in “Recent topics in Differential and Analytic Geometry” (T. Ochiai, ed.) Kinokuniya, 1990.

  13. R. Greene, Some concepts and methods in Riemannian geometry, in “Differential Geometry: Riemannian Geometry” (R. Greene, S.T. Yau, eds.), Proceedings of AMS Symposia in Pure Math. 54:3 (1993), 1–22.

    Google Scholar 

  14. M. Gromov, Volume and bounded cohomology, Publication IHES 56 (1982), 5–99.

    Google Scholar 

  15. M. Gromov, Asymptotic invariants of infinite groups, in “Geometric Group Theory” vol. 2 (G. Niblo, M. Roller, eds.) Cambridge University Press, 1993.

  16. M. Gromov, J. Lafontaine, P. Pansu, Structures metriques pour les varietes Riemannienes, CEDIC/Fernand Nathan, Paris, 1981.

    Google Scholar 

  17. M. Gromov, H.B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES 58 (1983), 295–408.

    Google Scholar 

  18. K. Grove, Metric differential geometry, in “Differential Geometry”, (V. Hansen, ed.), Springer Lecture Notes in Math. 1263 (1987), 171–227.

    Google Scholar 

  19. K. Grove, P. Petersen V, Bounding homotopy type by geometry, Ann. Math. 128 (1988), 195–206.

    Google Scholar 

  20. K. Grove, P. Petersen V, J. Wu, Geometric finiteness theorems via controlled topology, Inv. Math. 99 (1990), 205–213.

    Google Scholar 

  21. M. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72.

    Google Scholar 

  22. M. Kreck, S. Stolz, Non-connected moduli spaces of positive sectional curvature metrics, J. of the Amer. Math. Soc. 6:4 (1993), 825–850.

    Google Scholar 

  23. C.F. Miller, Decision problems for groups — survey and reflections, in “Combinatorial Group Theory” (G. Baumslag, C.F. Miller, eds.), Springer, 1989, 1–59.

  24. J. Milnor, Lectures on the h-cobordism Theorem, Princeton Univ. Press, 1965.

  25. A. Nabutovsky, Non-recursive functions in real algebraic geometry, Bull. Amer. Math. Soc. 20 (1989), 61–65.

    Google Scholar 

  26. A. Nabutovsky, Einstein structures: existence versus uniqueness, Geometric And Functional Analysis 5 (1995), 76–91.

    Google Scholar 

  27. A. Nabutovsky, Non-recursive functions, knots ‘with thick ropes’ and self-clenching ‘thick’ hypersurfaces, Comm. on Pure and Appl. Math. 48 (1995), 381–428.

    Google Scholar 

  28. A. Nabutovsky, Geometry of the space of triangulations of a compact manifold, Comm. Math. Phys., to appear.

  29. I. Nikolaev, Bounded curvature closure of the set of compact Riemannian manifolds, Bull. AMS 24 (1992), 171–177.

    Google Scholar 

  30. P. Pansu, Effondrement des varietes riemanniennes (d'apres J. Cheeger et M. Gromov), Sem. Bourbaki 1983/1984, Asterisque 121–122 (1985), 63–82.

  31. S. Peters, Convergence of Riemannian manifolds, Comp. Math. 62 (1987), 3–16.

    Google Scholar 

  32. P. Petersen V, Gromov-Hausdorff convergence of metric spaces, in “Differential Geometry: Riemannian Geometry” (R. Greene, S.T. Yau, ed.), Proceedings of AMS Symposia in Pure Math. 54:3 (1993), 489–504.

    Google Scholar 

  33. C. Plaut, Metric curvature, convergence and topological finiteness, Duke Math. J. 66:1 (1992), 43–57.

    Google Scholar 

  34. J.J. Rotman, An Introduction to the Theory of Groups, Allyn and Bacon, Boston, 1984.

    Google Scholar 

  35. A. Thompson, Thin position and the recognition problem forS 3, Math. Res. Lett. 1 (1994), 613–630.

    Google Scholar 

  36. I. Volodin, V. Kuznetzov, A. Fomenko, The problem of discriminating algorithmically the standard three-dimensional sphere, Russ. Math. Surveys 29:5 (1974), 71–172.

    Google Scholar 

  37. H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, 1957.

    Google Scholar 

  38. T. Yamaguchi, Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds, Proc. Amer. Math. Soc. 102 (1988), 660–666.

    Google Scholar 

Download references

Author information

Author notes

  1. Alexander Nabutovsky

    Present address: Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada

Authors and Affiliations

  1. Courant Institute of Math. Sci., New York University, 251 Mercer Street, 10012, New York, NY, USA

    Alexander Nabutovsky

Authors
  1. Alexander Nabutovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was partially supported by the New York University Research Challenge Fund grant, by NSF grant DMS 9114456 and by the NSERC operating grant OGP0155879.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nabutovsky, A. Disconnectedness of sublevel sets of some Riemannian functionals. Geometric and Functional Analysis 6, 703–725 (1996). https://doi.org/10.1007/BF02247118

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02247118

Keywords

Search

Navigation