  We consider minimal surfaces $M$ which are complete, embedded and have finite
total curvature in $\R^3$, and bounded, entire solutions with finite Morse
index of the Allen-Cahn equation $\Delta u + f(u) = 0 \hbox{in} \R^3 $. Here
$f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14
(1-u^2)^2$. We assume that $M$ has $m\ge 2$ ends, and additionally that $M$ is
non-degenerate, in the sense that its bounded Jacobi fields are all originated
from rigid motions (this is known for instance for a Catenoid and for the
Costa-Hoffman-Meeks surface of any genus). We prove that for any small $\alpha
>0$, the Allen-Cahn equation has a family of bounded solutions depending on
$m-1$ parameters distinct from rigid motions, whose level sets are embedded
surfaces lying close to the blown-up surface $M_\alpha := \alpha^{-1} M$, with
ends possibly diverging logarithmically from $M_\A$. We prove that these
solutions are $L^\infty$-{\em non-degenerate} up to rigid motions, and find
that their Morse index coincides with the index of the minimal surface. Our
construction suggests parallels of De Giorgi conjecture for general bounded
solutions of finite Morse index.
