  In this paper we prove two results. The first shows that the
Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can
determine its topological, differential, and metric structure. Earlier work of
this type assumes a priori that the surface is a planar domain [36] or that the
geometry is a priori known [29]. We will then apply this result to study a
geometric inverse problem for determining minimal surfaces embedded in
$3$-dimensional Riemannian manifolds. In particular we will show that knowledge
of the volumes of embedded minimal surfaces determine not only their
topological and differential structure but also their Riemannian structure as
an embedded hypersurface. Such geometric inverse problems are partially
inspired by the physical models proposed by the AdS/CFT correspondence.
  The crucial ingredient in removing the planar domain assumption is the
determination of the boundary trace of holomorphic functions from knowledge of
the Dirichlet-Neumann map of $\Delta_g +q$. This requires a new type of
argument involving Carleman estimates and construction of CGO whose phase
functions are not Morse as in the case of [29]. We anticipate that these
techniques could be of use for studying other inverse problems in geometry and
PDE.
