  Extensions of the Standard Model with $N$ Higgs doublets are simple
extensions presenting a rich mathematical structure. An underlying Minkowski
structure emerges from the study of both variable space and parameter space.
The former can be completely parametrized in terms of two future lightlike
Minkowski vectors with spatial parts forming an angle whose cosine is
$-(N-1)^{-1}$. For the parameter space, the Minkowski parametrization enables
one to impose sufficient conditions for bounded below potentials, characterize
certain classes of local minima and distinguish charge breaking vacua from
neutral vacua. A particular class of neutral minima presents a degenerate mass
spectrum for the physical charged Higgs bosons.
