  Measurements of ferroelectric polarization and dielectric constant were done
on $R$Mn$_2$O$_5$ ($R$=Tb, Dy, and Ho) with applied hydrostatic pressures of up
to 18 kbar. At ambient pressure, distinctive anomalies were observed in the
temperature profile of both physical properties at critical temperatures
marking the onset of long range AFM order (T$_{N1}$), ferroelectricity
(T$_{C1}$) as well as at temperatures when anomalous changes in the
polarization, dielectric constant and spin wave commensurability have been
previously reported. In particular, the step in the dielectric constant at low
temperatures (T$_{C2}$), associated with both a drop in the ferroelectric
polarization and an incommensurate magnetic structure, was shown to be suddenly
quenched upon passing an $R$-dependent critical pressure. This was shown to
correlate with the stabilization of the high ferroelectric polarization state
which is coincident with the commensurate magnetic structure. The observation
is suggested to be due to a pressure induced phase transition into a
commensurate magnetic structure as exemplified by the pressure-temperature
($p$-$T$) phase diagrams constructed in this work. The $p$-$T$ phase diagrams
are determined for all three compounds.
