A positive resolution of De Giorgi's conjecture in dimension eight

We resolve De Giorgi's conjecture positively in dimension eight: every entire $C^2$ solution $u:\R^8\to(-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v:\R^7\to[-1,1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.