On the asymptotic Plateau's problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Abstract

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-\alpha$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $\Gamma$ in the asymptotic boundary of $M,$ and $|H|<\sqrt{\alpha}$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $\Gamma$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symmetric spaces of non compact type and to more general boundary data.