  In this paper we extend and simplify previous results regarding the
computation of Euclidean Wilson loops in the context of the AdS/CFT
correspondence, or, equivalently, the problem of finding minimal area surfaces
in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary
curve X(s) we define, using the integrable properties of the system, a family
of curves X(lambda,s) depending on a complex parameter lambda known as the
spectral parameter. This family has remarkable properties. As a function of
lambda, X(lambda,s) has cuts and therefore is appropriately defined on a
hyperelliptic Riemann surface, namely it determines the spectral curve of the
problem. Moreover, X(lambda,s) has an essential singularity at the origin
lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0,
when appropriately integrated along the curve give the area of the
corresponding minimal area surface.
  Furthermore we show that the same construction allows the computation of
certain surfaces with one or more boundaries corresponding to Wilson loop
correlators. We extend the area formula for that case and give some concrete
examples. As the main example we consider a surface ending on two concentric
circles and show how the boundary circles can be deformed by introducing extra
cuts in the spectral curve.
