Fermat functional equations over Riemann surfaces

We investigate the existence of non-trivial holomorphic and meromorphic solutions of Fermat functional equations over an open Riemann surface $S$. When $S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always exists non-trivial holomorphic and meromorphic solution. When $S$ is a general open Riemann surface, we prove that every non-trivial holomorphic or meromorphic solution satisfies a growth condition, provided that the power exponents of the equations are bigger than some certain positive integers.