No abstract

In this article we introduce a new conformal invariant and we prove a conformal rigidity theorem which has no restriction on the size of the codimension. We also prove an isometric rigidity theorem whose assumptions are less restrictive than in Allendoerfer's theorem.Introduction.

In this paper, we make estimates for the radius of balls contained in some component of the complementary of a complete hypersurface into a space form, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The results are obtained using an algebraic lemma and a tangency principle related with the length of the second fundamental form. The algebraic lemma also is used to improve a result for graphs due to Hasanis-Vlachos. Mathematics Subject Classifications (2000). 53C42, 53C21, 35B50, 35J60.

In this paper we generalize and extend to any Riemannian manifold maximum principles for Euclidean hypersurfaces with vanishing curvature functions obtained by Hounie-Leite.

In this paper we prove a tangency principle (see Fontenele and Silva 2001) related with the length of the second fundamental form, for hypersurfaces of an arbitrary ambient space. As geometric applications, we make radius estimates of the balls that lie in some component of the complementary of a complete hypersurface into Euclidean space, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The basic tool established here is that some operator is elliptic at points where the second fundamental form is positive definite.