Complete self-shrinkers with constant norm of the second fundamental form in \(\RR^4\)

Let \(X:M^3\to\RR^4\) be a complete self-shrinker with constant squared norm \(S\) of the second fundamental form. We prove that \(S=0\) or \(S=1\), and that \(X(M)\) is isometric to a hyperplane or a standard product \(\SSS^k(\sqrt{k})\times\RR^{3-k}\), \(1\leq k\leq3\). This proves a conjecture pr