Consider for instance a parallel plate, a spherical or a cylindrical capacitor. Usually we analyze it by considering the surface charge density on each plate uniform, i.e. constant along the plate.
Why do we understand that? The only explanation I have found on the web is: "Because like charges repel one another".
Ok, I understand that, because of Coulomb interaction, all charges I put on a metal surface tend to separate themself in order to minimize that force, but I do not understand why should it be a uniform distribution.
Consider for instance a situation like this (reference), in which obviously the same principle ("Because like charges repel one another") is correct:
You see that charge density becomes most concentrated at the location of greatest curvature, it is a known effect. Charges repel themselves, but charge density is not uniform.
Well, I do not see differences with a plate of a capacitor: in both cases a charge is put on a conductor. In case of a parallel plate capacitor, it has a rectangular shape, in the last picture, it is like a warped circle. What will happen if I build a capacitor with two parallel plates with that shape?
So, I do not understand which is the connection between the capacitor geometry and surface charge density (it seems that "nice geometry" means uniform distribution...).