  It is shown that in the Einstein-conformally coupled Higgs--Maxwell system
with Friedman-Robertson-Walker symmetries the energy density of the Higgs field
has stable local minimum only if the mean curvature of the $t={\rm const}$
hypersurfaces is less than a finite critical value $\chi_c$, while for greater
mean curvature the energy density is not bounded from below. Therefore, there
are extreme gravitational situations in which even quasi-locally defined
instantaneous vacuum states of the Higgs sector cannot exist, and hence one
cannot at all define the rest mass of all the classical fields. On
hypersurfaces with mean curvature less than $\chi_c$ the energy density has the
`wine bottle' (rather than the familiar `Mexican hat') shape, and the gauge
field can get rest mass via the Brout--Englert--Higgs mechanism. The spacelike
hypersurface with the critical mean curvature represents the moment of
`genesis' of rest masses.
