Minkowski space structure of the Higgs potential in 2HDM
  The Higgs potential of 2HDM keeps its generic form under the group of transformation GL(2,C), which is larger than the usually considered reparametrization group U(2). This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly CP-conserving Higgs potential has spontaneously CP-violating minima. Our analysis avoids manipulation with high-order algebraic equations. 
I. P. Ivanov