Any holomorphic function on a compact Riemann surface is necessarily constant. , so it is constant, by the maximum modulus principle.

How do you show that a complex function is constant?

is analytic in C then it is a constant function. This can only happen when ux and uy are equal 0. As the partial derivatives of f are 0, f must be a constant function.

When an entire function is constant?

Liouville's theorem states that any bounded entire function must be constant. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.