Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and \(S\le 1\)

Let \(X:M^n\to\mathbb R^{n+p}\) be an \(n\)-dimensional complete self-shrinker in arbitrary codimension \(p\ge 1\). Suppose that the scalar curvature \(R\) is a positive constant and that the squared norm \(S\) of the second fundamental form satisfies \(S\le 1\). We prove that \(S\equiv 1\) and that