I'm struggling to understand why the maximum modulus principle holds. Why is it that a holomorphic function that is nonconstant in an open subset of $\mathbb{C}$ cannot have a local maximum inside the subset? Intuitively, why are no such curves with local maxima not holomorphic?
More specifically, I've been trying to figure out why a holomorphic function on compact complex connected curve must be constant. This directly follows from the maximum modulus principle because there must be a maximum due to compactness, but I don't understand why either of these properties hold.
Best Answer
If $f$ is holomorphic, it's real and imaginary part are harmonic, i.e. they satisfy Laplace's equation. This is the equation satisfied by a stretched membranes. Such a physical configuration intuitively cannot have bumps. That is a cool way to see it.