  The measured values of the Higgs and top quark masses imply that the Standard
Model potential is very likely to be unstable at large Higgs values. This is
particularly problematic during inflation, which sources large perturbations of
the Higgs. The instability could be cured by a threshold effect induced by a
scalar with a large vacuum expectation value and directly connected to the
Standard Model through a Higgs portal coupling. However, we find that in a
minimal model in which the scalar generates inflation, this mechanism does not
stabilize the potential because the mass required for inflation is beyond the
instability scale. This conclusion does not change if the Higgs has a direct
weak coupling to the scalar curvature. On the other hand, if the potential is
absolutely stable, successful inflation in agreement with current CMB data can
occur along a valley of the potential with a Mexican hat profile. We revisit
the stability conditions, independently of inflation, and clarify that the
threshold effect cannot work if the Higgs portal coupling is too small. We also
show that inflation in a false Higgs vacuum appearing radiatively for a tuned
ratio of the Higgs and top masses leads to an amplitude of primordial
gravitational waves that is far too high, ruling out this possibility.
