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Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subsetof $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convexhull. In this paper we make use of the classical Minkowski problem, to show thecomplete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundarysurface 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 homeomorphismonto $\mathbb{S}^2-\{p_j\}_{j=1}^m$ and $D_j\bot p_j,$ for all $j.$  We derive applications to the generalized Minkowski problem, existence ofharmonic diffeomorphisms between domains of $\mathbb{S}^2,$ existence ofcapillary surfaces in $\mathbb{R}^3,$ and a Hessian equation of Monge-Amperetype.

The Minkowski problem, new constant curvature surfaces in R^3, and some applications