The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

A Bonnet theorem for isometric immersions into products of space forms

We prove a Bonnet theorem for isometric immersions of semi-Riemannian manifolds into products of semi-Riemannian space forms. Namely, we give necessary and sufficient conditions for the existence and uniqueness (up to an isometry of the ambient space) of an isometric immersion of a semi-Riemannian manifold into a product of semi-Riemannian space forms.     

On submanifolds with zero normal torsion in Euclidean space

The research was supported by the International Foundation A Cultural Initiative and the Russian Academy of Natural Sciences.     

On the first Hodge eigenvalue of isometric immersions

We give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on -forms on a compact manifold without boundary isometrically immersed in or . The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature.

     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.