  We study the Dirichlet problem for the following prescribed mean curvature
PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla
v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega.
\end{cases} $$ where $\Omega$ is a domain contained in a complete Riemannian
manifold $M,$ $f:\Omega\times\mathbb{R\rightarrow R}$ is a fixed function and
$\varphi$ is a given continuous function on $\partial\Omega$. This is done in
three parts. In the first one we consider this problem in the most general
form, proving the existence of solutions when $\Omega$ is a bounded
$C^{2,\alpha}$ domain, under suitable conditions on $f$, with no restrictions
on $M$ besides completeness. In the second part we study the asymptotic
Dirichlet problem when $M$ is the hyperbolic space $\mathbb{H}^n$ and $\Omega$
is the whole space. This part uses in an essential way the geometric structure
of $\mathbb{H}^n$ to construct special barriers which resemble the Scherk type
solutions of the minimal surface PDE. In the third part one uses these Scherk
type graphs to prove the non existence of isolated asymptotic boundary
singularities for global solutions of this Dirichlet problem.

Differential Geometry