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

Abstract

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 $${f : \Sigma^{n} \rightarrow \mathbb{H}^{n+1}}$$ 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.