  Little seems to be known about the invariant manifolds for stochastic partial
differential equations (SPDEs) driven by nonlinear multiplicative noise. Here
we contribute to this aspect and analyze the Lu-Schmalfu{\ss} conjecture
[Garrido-Atienza, et al., J. Differential Equations, 248(7):1637--1667, 2010]
on the existence of stable manifolds for a class of parabolic SPDEs driven by
nonlinear mutiplicative fractional noise. We emphasize that stable manifolds
for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron
method cannot be applied, since the Lyapunov-Perron operator does not give any
information about the backward orbit. However, by means of interpolation
theory, we construct a suitable function space in which the discretized
Lyapunov-Perron-type operator has a unique fixed point. Based on this we
further prove the existence and smoothness of local stable manifolds for such
SPDEs.
