  We consider a one-dimensional fluctuating interfacial profile governed by the
Edwards-Wilkinson or the stochastic Mullins-Herring equation for periodic,
standard Dirichlet and Dirichlet no-flux boundary conditions. The minimum
action path of an interfacial fluctuation conditioned to reach a given maximum
height $M$ at a finite (first-passage) time $T$ is calculated within the
weak-noise approximation. Dynamic and static scaling functions for the profile
shape are obtained in the transient and the equilibrium regime, i.e., for
first-passage times $T$ smaller or lager than the characteristic relaxation
time, respectively. In both regimes, the profile approaches the maximum height
$M$ with a universal algebraic time dependence characterized solely by the
dynamic exponent of the model. It is shown that, in the equilibrium regime, the
spatial shape of the profile depends sensitively on boundary conditions and
conservation laws, but it is essentially independent of them in the transient
regime.
