  What happens when the time evolution of a fluctuating interface is
interrupted with resetting to a given initial configuration after random time
intervals $\tau$ distributed as a power-law $\sim \tau^{-(1+\alpha)};~\alpha >
0$? For an interface of length $L$ in one dimension, and an initial flat
configuration, we show that depending on $\alpha$, the dynamics as $L \to
\infty$ exhibits a rich long-time behavior. Without resetting, the interface
width grows unbounded with time as $t^\beta$, where $\beta$ is the so-called
growth exponent. We show that introducing resetting induces for $\alpha>1$ and
at long times fluctuations that are bounded in time. Corresponding to such a
stationary state is a distribution of fluctuations that is strongly
non-Gaussian, with tails decaying as a power-law. The distribution exhibits a
cusp for small argument, implying that the stationary state is out of
equilibrium. For $\alpha<1$, resetting is unable to counter the otherwise
unbounded growth of fluctuations in time, so that the distribution of
fluctuations remains time dependent with an ever-increasing width even at long
times. Although stationary for $\alpha>1$, the width of the interface grows
forever with time as a power-law for $1<\alpha < \alpha^{({\rm w})}$, and
converges to a finite constant only for larger $\alpha$, thereby exhibiting a
crossover at $\alpha^{({\rm w})}=1+2\beta$. The time-dependent distribution of
fluctuations for $\alpha<1$ exhibits for small argument another interesting
crossover behavior, from cusp to divergence, across $\alpha^{({\rm
d})}=1-\beta$. We demonstrate these results by exact analytical results for the
paradigmatic Edwards-Wilkinson (EW) dynamical evolution of the interface, and
further corroborate our findings by extensive numerical simulations of
interface models in the EW and the Kardar-Parisi-Zhang universality class.
