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 $u:{\mathbb{R}}^8\to (-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v:{\mathbb{R}}^7\to [-1,1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.