Illinois Journal of Mathematics

A tangency principle and applications

F. Fontenele and Sérgio L. Silva

Full-text: Open access

Abstract

In this paper we obtain a tangency principle for hypersurfaces, with not necessarily constant $r$-mean curvature function $H_r $, of an arbitrary Riemannian manifold. That is, we obtain sufficient geometric conditions for two submanifolds of a Riemannian manifold to coincide, as a set, in a neighborhood of a tangency point. As applications of our tangency principle, we obtain, under certain conditions on the function $H_r$, sharp estimates on the size of the greatest ball that fits inside a connected compact hypersurface embedded in a space form of constant sectional curvature $c\leq 0$ and on the size of the smallest ball that encloses the image of an immersion of a compact Riemannian manifold into a Riemannian manifold with sectional curvatures limited from above. This generalizes results of Koutroufiotis, Coghlan-Itokawa, Pui-Fai Leung, Vlachos and Markvorsen. We also generalize a result of Serrin. Our techniques permit us to extend results of Hounie-Leite.

Article information

Source
Illinois J. Math., Volume 45, Number 1 (2001), 213-228.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138264

Digital Object Identifier
doi:10.1215/ijm/1258138264

Mathematical Reviews number (MathSciNet)
MR1849995

Zentralblatt MATH identifier
0987.53026

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 35B50: Maximum principles 35J60: Nonlinear elliptic equations 53C24: Rigidity results 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Fontenele, F.; Silva, Sérgio L. A tangency principle and applications. Illinois J. Math. 45 (2001), no. 1, 213--228. doi:10.1215/ijm/1258138264. https://projecteuclid.org/euclid.ijm/1258138264


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