Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T21:12:56.769Z Has data issue: false hasContentIssue false
  • English
  • Français

The minimal volume of the plane

Part of: Global differential geometry Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

B. H. Bowditch
B. H. Bowditch
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton, SO9 5NH, Great Britain, email: bhb@maths.soton.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give an account of the minimal volume of the plane, as defined by Gromov, and first computed by Bavard and Pansu. We also describe some related geometric inequalities.

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 1 , August 1993 , pp. 23 - 40
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bavard, C. and Pansu, P., ‘Sur le volume minimal de R2’, Ann. Scient. Éc. Norm. Sup. 19 (1986), 479–490.Google Scholar
[2]Bowditch, B. H., ‘Notes on locally CAT(1)spaces’, preprint, Aberdeen, 1992.Google Scholar
[3]Charney, R. and Davis, M., ‘Singular metrics of non-positive curvature on branched covers of riemannian manifolds’, preprint, Ohio State, 1990.Google Scholar
[4]Cheeger, J. and Ebin, D. G., Comparison theorems in riemannian geometry (North-Holland, Amsterdam, 1975).Google Scholar
[5]Cheeger, J. and Gromov, M., ‘Collapsing riemannian manifolds while keeping their curvature bounded I’, J. Differential Geom. 23 (1986), 309–346.Google Scholar
[6]Cheeger, J. and Gromov, M., ‘Collapsing riemannian manifolds while keeping their curvature bounded II’, J. Differential Geom. 32 (1990), 269–298.Google Scholar
[7]Gallot, S., ‘Volume minimal des variétés hyperboliques: un théorème local et un resultat global’, in: Séminaire de théorie spectrale et géométrie, vol. 7 (Grenoble, St. Martin d' Hères, 1988) pp. 35–52.Google Scholar
[8]Gromov, M., ‘Volume and bounded cohomology’, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.Google Scholar
[9]Hocking, J. G. and Young, G. S., Topology (Addison-Wesley, Reading, 1961).Google Scholar
[10]Jaco, W. H. and Shalen, P. B., ‘Seifert fibered spaces in 3-manifolds’, Amer. Math. Soc. Mem. 220 (1979).Google Scholar
[11]Johannson, K., Homotopy equivalences of 3-manifolds with boundary, Springer Lecture Notes in Mathematics 761 (Springer-Verlag, Berlin, 1979).Google Scholar
[12]Klingenberg, W., Riemannian geometry, de Gruyter Studies in Mathematics 1 (de Gruyter, Amsterdam, 1982).Google Scholar
[13]Moise, E. E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics 47 (Springer, Berlin, 1977).CrossRefGoogle Scholar
[14]Osserman, R., ‘Bonnesen-style inequalities’, Amer. Math. Monthly 86 (1977), 1–29.Google Scholar
[15]Osserman, R., ‘The isoperimetric inequality’, Bull. Amer. Math. Soc. 84 (1978), 1182–1238.CrossRefGoogle Scholar
[16]Thurston, W. P., ‘Three dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. 6 (1982), 357–381.Google Scholar
[17]Yang, D., ‘Convergence of riemannian manifolds with integral bounds on curvature I’, Ann. Scient. Éc. Norm. Sup. 25 (1992), 77–105.Google Scholar