Classification without the bound on the squared second fundamental form

Classify complete self-shrinkers in arbitrary codimension with positive constant scalar curvature after removing the assumption that the squared norm S of the second fundamental form satisfies S≤1, including the higher-codimension product examples with S>1.

Background

The paper proves that every complete self-shrinker in arbitrary codimension with positive constant scalar curvature and S≤1 is isometric to a round sphere or a standard generalized cylinder. The restriction S≤1 is used to invoke the Cao–Li gap theorem. The authors note that higher-codimension products of round spheres and Euclidean factors provide examples with positive constant scalar curvature and S>1, so removing the bound requires a broader classification encompassing these examples.