Tohoku Mathematical Journal
2004
June
SECOND SERIES VOL. 56, NO. 2

Tohoku Math. J.
56 (2004), 155-162

Title ON STABLE COMPLETE HYPERSURFACES WITH VANISHING $r$-MEAN CURVATURE

Author Manfredo do Carmo and Maria F. Elbert

(Received July 10, 2002, revised August 22, 2003)
Abstract. A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no complete stable hypersurface of an $n$-euclidean space with vanishing $(n-1)$-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immersion is proper and the total curvature is finite.

2000 Mathematics Subject Classification. Primary 53C42.
Key words and phrases. Stability, $r$-mean curvature, complete, finite total curvature.

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