zbMATH — the first resource for mathematics

Estimates of volume by the length of shortest closed geodesics on a convex hypersurface. (English) Zbl 0563.53035
The author estimates the length L of a closed geodesic of minimal length on a closed convex hypersurface \(M^ n\) as follows \[ L^ n\leq (2\pi)^{n-1} \Gamma^ 2(n/2+1/2)Vol(M^ n). \] The main theorem is more general and applies to minimal k-spheres in \(M^ n\) if the usual minimax argument gives a solution - which has only been shown to be the case for \(k=2\) and \(n=3\).
Reviewer: G.Thorbergsson

53C22 Geodesics in global differential geometry
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
Full Text: DOI EuDML
[1] [B] Busemann, H.: Convex surfaces. New York: Interscience, 1958 · Zbl 0196.55101
[2] [C] Croke, C.: Poincaré’s Problem and the length of the shortest closed geodesic on a convex hypersurface. J. Differ. Geom.17, 595-634 (1982) · Zbl 0501.53031
[3] [G1] Gromov, M.: Structures metriques pour les variétiés riemanniennes, p. 47. Paris: CEDIC 1981
[4] [G2] Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom.18, 1-147 (1983) · Zbl 0515.53037
[5] [S] Santalo, L.: Integral geometry and geometric probability. Reading: Addison-Wesley, 1976
[6] [Sm] Smith, F.R.: On the existence of embedded minimal 2-spheres in the 3-sphere endowed with an arbitrary metric. Thesis, Melbourne (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.