  We investigate the UV-completion of the Higgs inflation in the metric and the
Palatini formalisms. It is known that the cutoff scales for the perturbative
unitarity of these inflation models become much smaller than the Planck scale
to be consistent with observations. Expecting that the low cutoff scale
originates in the curvature of a field-space spanned by the Higgs fields, we
consider embedding the curved field-space into a higher dimensional flat space
and apply this procedure to the metric-Higgs and the Palatini-Higgs scenarios.
The new field introduced in this way successfully flattens the field-space and
UV-completes the Higgs inflation in the metric formalism. However, in the
Palatini formalism, the new field cannot uplift the cutoff up to the Planck
scale. We also discuss the unavoidable low cutoff in the Palatini formalism in
the context of the local conformal symmetry.
