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 $X$ is isometric to either the round sphere $S^n(\sqrt{n})$ or the standard generalized cylinder $S^k(\sqrt{k})\times {\mathbb{R}}^{n-k}$, $2\le k\le n-1$. In particular, no new higher-codimension examples occur under these assumptions. The proof uses an algebraic lemma which yields a uniform positive lower bound for the Bakry–Émery Ricci curvature, together with the comparison theorem of Wei–Wylie, the equivalence of finite Gaussian volume and polynomial volume growth due to Cheng–Zhou, and the gap theorem of Cao–Li.