  We consider n-dimensional deterministic flows obtained by perturbing a
gradient flow. We assume that the gradient flow admits a stable curve of
stationary points, and thus if the perturbation is not too large the perturbed
flow also admits an attracting curve. We show that the noise induced escape
problem from a stable fixed point of this curve can be reduced to a
one-dimensional problem: we can approximate the associated quasipotential by
the one associated to the restricted dynamics on the stable curve. The error of
this approximation is given in terms of the size of the perturbation.
