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.
References
Can the assumption S\le 1 be removed? As pointed out in Remark \ref{remark-S}, in higher codimension there exist product examples with S>1 and positive constant scalar curvature, so a complete classification must include such examples.
Can one classify complete self-shrinkers with negative or sign-changing scalar curvature? The algebraic Lemma \ref{alg} uses the positivity of R in an essential way, and the negative case requires a different approach.
Based on the above results, Cheng and Wei [10] proposed the following conjecture: Conjecture 1.1. An n-dimensional complete self-shrinker X : M n → Rn+1 with constant squared norm of the second fundamental form is isometric to one of a hyperplane Rn passing through the origin, a round sphere Sn(√n) with the center at the origin, or a cylinder Sk(√k) × Rn−k, 1 ≤ k ≤ n. In particular, S must be 0 or 1.