  Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset
of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex
hull. In this paper we make use of the classical Minkowski problem, to show the
complete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundary
surface consists of an open surface $S$ with constant Gauss curvature
(respectively, constant mean curvature) and $m$ planar compact discs
$\bar{D_1},...,\bar{D_m},$ such that the Gauss map of $S$ is a homeomorphism
onto $\mathbb{S}^2-\{p_j\}_{j=1}^m$ and $D_j\bot p_j,$ for all $j.$
  We derive applications to the generalized Minkowski problem, existence of
harmonic diffeomorphisms between domains of $\mathbb{S}^2,$ existence of
capillary surfaces in $\mathbb{R}^3,$ and a Hessian equation of Monge-Ampere
type.
