A celebrated conjecture due to De Giorgi states that any bounded
solution of the Allen-Cahn
  equation $\Delta u + (1-u2) u = 0 \  \hbox{in} \ R^N $
with  $\frac{\partial u}{\partial y_N} >0$ must be such that its level sets
$\{u=\lambda\}$ are all  hyperplanes, {\em \bf at least} for dimension $N\leq
8$. Based on a minimal graph $\Gamma$ which is not a hyperplane, found by
Bombieri,  De Giorgi and Giusti in $R^N$, $N\geq 9$, we prove  that for any
small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with
$\pp_{y_N}u_\alpha >0$, which resembles
 $ \tanh \left ( \frac t{\sqrt{2}}\right ) $,
where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph
$\Gamma_\alpha := \alpha^{-1}\Gamma$.
This solution constitutes a counterexample to De Giorgi conjecture for $N\geq
9$. The methods and techniques are extended to establish a correspondence
between   minimal
surfaces $M$ which are complete, embedded and have finite total curvature in
$\R3$, and
bounded, entire solutions with finite Morse index of  the Allen-Cahn equation.
 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. (Joint work with M. del Pino and M Kowalczyk)