On Frankel’s Theorem

Peter Petersen, Frederick Wilhelm

Type: Article

Publication Date: 2003-03-01

Citations: 26

DOI: https://doi.org/10.4153/cmb-2003-013-4

Abstract

Abstract In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

Locations

Canadian Mathematical Bulletin - View - PDF

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Works Cited by This (9)

Action	Title	Year	Authors
+ PDF Chat	The splitting theorem for manifolds of nonnegative Ricci curvature	1971	Jeff Cheeger Detlef Gromoll
+ PDF Chat	Integral curvature bounds, distance estimates and applications	1998	Peter Petersen Chadwick Sprouse
+ PDF Chat	Proof of the soul conjecture of Cheeger and Gromoll	1994	G. Perelman
+	A warped product splitting theorem	1992	Christopher B. Croke Bruce Kleiner
+	An extension of E. Hopf’s maximum principle with an application to Riemannian geometry	1958	Eugenio Calabi
+ PDF Chat	Manifolds with positive curvature	1961	Theodore Frankel
+ PDF Chat	Maximum principle for hypersurfaces	1989	J.-H. Eschenburg
+ PDF Chat	On the Structure of Complete Manifolds of Nonnegative Curvature	1972	Jeff Cheeger Detlef Gromoll
+	Riemannian geometry	1987	Sylvestre Gallot