  The main goal of this article is to understand the trace properties of
nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a
graphical structure.
  We establish that at any boundary points at which the trace from inside
happens to coincide with the exterior datum, also the tangent planes of the
traces necessarily coincide with those of the exterior datum.
  This very rigid geometric constraint is in sharp contrast with the case of
the solutions of the linear equations driven by the fractional Laplacian, since
we also show that, in this case, the fractional normal derivative can be
prescribed arbitrarily, up to a small error.
  We remark that, at a formal level, the linearization of the trace of a
nonlocal minimal graph is given by the fractional normal derivative of a
fractional Laplace problem, therefore the two problems are formally related.
Nevertheless, the nonlinear equations of fractional mean curvature type present
very specific properties which are strikingly different from those of other
problems of fractional type which are apparently similar, but diverse in
structure, and the nonlinear case given by the nonlocal minimal graphs turns
out to be significantly more rigid than its linear counterpart.
