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The AdS/CFT correspondence relates Wilson loops in $N$=4 SYM theory tominimal area surfaces in AdS space. If the loop is a plane curve the minimalsurface lives in hyperbolic space $H_3$ (or equivalently Euclidean AdS$_3$space). We argue that finding the area of such extremal surface can be easilydone 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 anyvalue of $\lambda$. This problem is equivalent to the statement that themonodromy matrix is trivial. It can be restated as that of finding a onecomplex parameter family of curves $X(\lambda,s)$ where $X(\lambda=1,s)$ is thegiven shape of the Wilson loop and such that the Schwarzian derivative$\{X(\lambda,s),s\}$ is meromorphic in $\lambda$ with only two simple poles. Wepresent a formula for the area in terms of the functions $V_{0,1,2}$ anddiscuss solutions to these equivalent problems in terms of theta functions.Finally, we also consider the near circular Wilson loop clarifying itsintegrability properties and rederiving its area using the methods described inthis paper.

Wilson loops and minimal area surfaces in hyperbolic space