Why does the maximum modulus principle hold intuitively?

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.

• It is a consequence of the “open mapping theorem” (which you might find more intuitive or not :) – Martin R Jan 12 '20 at 21:41

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.
• if $u=Re f$, then $\nabla^2 u=0$ (Laplace's equation, en.wikipedia.org/wiki/Laplace%27s_equation). This is nothing more than a rewriting of Cauchy-Riemann conditions. And it is the equation that an actual physical membrane follows. For instance consider a drum, by membrane I mean thin layer of material you hit to produce sound. if you imagine for instance chopping off a bit of the side of the drum, the resting form of the membrane will be the solution of Laplace's equation. – David Jaramillo Jan 12 '20 at 21:50