Francisco Fontenele

The length of the second fundamental form, a tangency principle and applications

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.

Maximum principles for hypersurfaces with vanishing curvature functions in an arbitrary Riemannian manifold

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

Heinz type estimates for graphs in Euclidean space

Let M n be an entire graph in the Euclidean (n+1)-space ℝ n+1 . Denote by H, R and |A|, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of M n . We prove that if the mean curvature H of M n is bounded, then inf M |R| = 0, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of M n is negative, then inf M |A| = 0. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate inf |H|, inf |R| and inf |A| for graphs in ℝ n+1 of C 2 real-valued functions defined on closed balls in ℝ n .

A characterization of round spheres in space forms

Let

A tangency principle and applications

\mathbb Q^{n+1}_c

Sharp Estimates for the Size of Balls in the Complement of a Hypersurface

be the complete simply-connected

Good Shadows, Dynamics and Convex Hulls of Complete Submanifolds

(n+1)

A Riemannian Bieberbach estimate

-dimensional space form of curvature

Good shadows, dynamics, and convex hulls

c

On the complexity of isometric immersions of hyperbolic spaces in any codimension

. In this paper we obtain a new characterization of geodesic spheres in