Indeed, it is the argmax of the likelihood, but the likelihood is not invariant under coordinate transformations. The quantity p(x)dx is invariant, not p(x). By

Indeed, it is the argmax of the likelihood, but the likelihood is not invariant under coordinate transformations. The quantity p(x)dx is invariant, not p(x). By picking a suitable coordinate transformation you can put the MLE on any value where the likelihood is not zero.