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Sociedad Colombiana de Matemáticas:Publicaciones

Revista Colombiana de Matemáticas

Volumen 36 [2] (2002)Páginas 97 - 106

Stable minimal cones in R^8 and R^9 with constant scalar curvature

Oscar Perdomo
Universidad del Valle, Cali, COLOMBIA

Abstract. In this paper we prove that if M Rn, n = 8 or n = 9, is a n - 1 dimensional stable minimal complete cone such that its scalar curvature varies radially, then M must be either a hyperplane or a Clifford minimal cone. By Gauss' formula, the condition on the scalar curvature is equivalent to the condition that the function 1(m)2 + · · · + n-1(m)2 varies radially. Here the i are the principal curvatures at m M . Under the same hypothesis, for M R10 we prove that if not only 1(m)2 + · · · + n-1(m)2 varies radially but either 1(m)3 + · · · + n-1(m)3 varies radially or 1(m)4 + · · · + n-1(m)4 varies radially, then M must be either a hyperplane or a Clifford minimal cone.

Palabras claves. Clifford hypersurfaces, minimal hypersurfaces, shape operator.

Codigo AMS. 2000 Mathematics Subject Classification. Primary: 53C42. Secondary: 58J50.

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