  In this paper we consider an inverse problem of determining a minimal surface
embedded in a Riemannian manifold. We show under a topological condition that
if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge
of the Dirichlet-to-Neumann map associated to the minimal surface equation
determines $\Sigma$ up to an isometry. Without the topological condition, we
show that a conformal factor of a general minimal surface $\Sigma$ can be
recovered.
  We develop a semiclassical nonlinear calculus for complex geometric optics
solutions, which allows an efficient error analysis for multiplication of the
correction terms of the solutions. The calculus is independent of the
application to the minimal surface equation and we expect it to have
applications in various inverse problems for nonlinear equations in dimension
$2$, in both $\mathbb{R}^2$ and geometric settings. Other applications of the
results include generalized boundary rigidity problem and the AdS/CFT
correspondence in physics.
