The generalized Chebotarev problem in higher genus

We consider the extension to higher genus Riemann surfaces of the classical Chebotarev problem, with a view towards the development of the theory of Pad\'e\ approximants on algebraic curves. To this end we define an appropriate notion of capacity that mimics the standard one, following works of Chirka and of the author and collaborators. The nontrivial topology of the Riemann surface requires further specification of the ``homotopy class'' of the continua in the solution of the Chebotarev problem. We also discuss the relationship of this problem to the theory of Jenkins-Strebel quadratic differentials.