A Classification of Complete Self-shrinkers

Let $X:M^n\to\R^{n+1}$ be an $n$-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere $S^n(\sqrt n)$ and the standard generalized cylinder $S^k(\sqrt k)\times \R^{n-k