  We report the results of a $J$-band survey for photometric variability in a
sample of young, low-gravity objects using the New Technology Telescope (NTT)
and the United Kingdom InfraRed Telescope (UKIRT). Surface gravity is a key
parameter in the atmospheric properties of brown dwarfs and this is the first
large survey that aims to test the gravity dependence of variability
properties. We do a full analysis of the spectral signatures of youth and
assess the group membership probability of each target using membership tools
from the literature. This results in a 30 object sample of young low-gravity
brown dwarfs. Since we are lacking in objects with spectral types later than
L9, we focus our statistical analysis on the L0-L8.5 objects. We find that the
variability occurrence rate of L0-L8.5 low-gravity brown dwarfs in this survey
is $30^{+16}_{-8}\%$. We reanalyse the results of Radigan 2014 and find that
the field dwarfs with spectral types L0-L8.5 have a variability occurrence rate
of $11^{+13}_{-4}\%$. We determine a probability of $98\%$ that the samples are
drawn from different distributions. This is the first quantitative indication
that the low-gravity objects are more likely to be variable than the field
dwarf population. Furthermore, we present follow-up $J_S$ and $K_S$
observations of the young, planetary-mass variable object PSO 318.5-22 over
three consecutive nights. We find no evidence of phase shifts between the $J_S$
and $K_S$ bands and find higher $J_S$ amplitudes. We use the $J_S$ lightcurves
to measure a rotational period of $8.45\pm0.05~$hr for PSO 318.5-22.
