Arch. Math. 93 (2009), 67–76 c  2009 Birkh¨auser Verlag Basel/Switzerland 0003-889X/09/010067-10 published online July 7, 2009 DOI 10.1007/s00013-009-0012-9 Archiv der Mathematik The influence of the boundary behavior on isometric immersions in the hyperbolic space L. Jorge, H. Mirandola, and F. Vitorio Abstract. This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infin- ity. 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 curva- ture 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 :Σ n → 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 immer- sions of arbitrary codimension. Mathematics Subject Classification (2000). Primary 53C42; Secondary 53C40. Keywords. Mean curvature, Hyperbolic space, Proper immersion, Asymptotic boundary, Tangency principle. 1. Introduction. Several works motivated principally by the Alexandrov reflection method showed that a properly embedded hypersurface into the hyperbolic space with constant mean curvature inherits the symmetry of its boundary (see [1, 2, 8, 9]). The m-dimensional hyperbolic space H m carries a natural compactification: H m = H m ∪ S m-1 (∞) where S m-1 (∞) is identified with the asymptotic classes of geodesic rays in H m and carries, in a natural way, the standard conformal structure (isometries of H m become conformal automorphisms of S m-1 (∞)). The asymptotic boundary This work is partially supported by CAPES, Brazil.