Illinois Journal of Mathematics

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin

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Abstract

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature $\ge\kappa$ is an Alexandrov's space of curvature $\ge\kappa$. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 1031-1033.

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403729

Digital Object Identifier
doi:10.1215/ijm/1254403729

Mathematical Reviews number (MathSciNet)
MR2546022

Zentralblatt MATH identifier
1200.53040

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53B25: Local submanifolds [See also 53C40]
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)

Citation

Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton. An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds. Illinois J. Math. 52 (2008), no. 3, 1031--1033. doi:10.1215/ijm/1254403729. https://projecteuclid.org/euclid.ijm/1254403729


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References

  • S. Buyalo, Shortest paths on convex hypersurface of a Riemannian manifold (Russian), Studies in Topology, Zap. Nauchn. Sem. LOMI 66 (1976), 114–132; translated in J. of Soviet Math. 12 (1979), 73–85.
  • R. E. Greene and H.-H. Wu, On the subharmonicity and plurisubharmonicity of geodesically convex functions, Indiana Univ. Math. J. 22 (1972/1973), 641–653.
  • A. D. Milka, Shortest arcs on convex surfaces (Russian), Dokl. Akad. Nauk SSSR 248 (1979), 34–36; translated in Soviet Math. Dokl. 20 (1979), 949–952.
  • P. Petersen, Riemannian geometry, Springer, New York, 1998.
  • A. Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203–219.