  The AdS/CFT correspondence relates Wilson loops in $N$=4 SYM theory to
minimal area surfaces in AdS space. If the loop is a plane curve the minimal
surface lives in hyperbolic space $H_3$ (or equivalently Euclidean AdS$_3$
space). We argue that finding the area of such extremal surface can be easily
done if we solve the following problem: given two real periodic functions
$V_{0,1}(s)$, $V_{0,1}(s+2\pi)=V_{0,1}(s)$, a third periodic function $V_2(s)$
is to be found such that all solutions to the equation $- \phi"(s) +
\big[V_0+{1\over 2} (\lambda+{1 \over \lambda}) V_1 + {i\over 2} (\lambda-{1
\over \lambda}) V_2\big] \phi(s)=0$ are anti-periodic in $s\in[0,2\pi]$ for any
value of $\lambda$. This problem is equivalent to the statement that the
monodromy matrix is trivial. It can be restated as that of finding a one
complex parameter family of curves $X(\lambda,s)$ where $X(\lambda=1,s)$ is the
given shape of the Wilson loop and such that the Schwarzian derivative
$\{X(\lambda,s),s\}$ is meromorphic in $\lambda$ with only two simple poles. We
present a formula for the area in terms of the functions $V_{0,1,2}$ and
discuss solutions to these equivalent problems in terms of theta functions.
Finally, we also consider the near circular Wilson loop clarifying its
integrability properties and rederiving its area using the methods described in
this paper.
