A celebrated conjecture due to De Giorgi states that any bounded

solution of the Allen-Cahn

equation $\Delta u + (1-u^{2}) 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

$\R^{3}$, 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)