Why is it that the field configuration that minimizes the Higgs potential is said to be a constant one (meaning $\phi(x) = \phi_0$)? Is it a physical reasoning, which kind of makes sense, since a non flat ground state of the field would be rather strange? Or is it a mathematical reasoning that the field configuration which minimizes the potential must be a constant one? If so how is it done? (It is not obvious to me right now, since the fields are functions of space-time (no quantization yet)).
I have read this and links therein but it did not really answered my question.