  A well-known question in classical differential geometry and geometric
analysis asks for a description of possible boundaries of $K$-surfaces, which
are smooth, compact hypersurfaces in $\mathbb{R}^d$ having constant Gauss
curvature equal to $K \geq 0$. This question generated a considerable amount of
remarkable results in the last few decades. Motivated by these developments
here we study the question of determining a $K$-surface when only part of its
boundary is fixed, and in addition the surface hits a given manifold at some
fixed angle. While this general setting is out of reach for us at the present,
we settle a model case of the problem, which in its analytic formulation
reduces to a Bernoulli type free boundary problem for the Monge-Amp\`ere
equation. We study both the cases of 0-curvature and of positive curvature. The
formulation of the free boundary condition and its regularity are the most
delicate and challenging questions addressed in this work. In this regard we
introduce a notion of a Blaschke extension of a solution which might be of
independent interest.
  The problem we study can also be interpreted as the Alt-Caffarelli problem
for the Monge-Amp\`ere equation. Moreover, it also relates to the problem of
isometric embedding of a positive metric on the annulus with partially
prescribed boundary and optimal transport with free mass.
