Estimates on scalar curvature of self-shrinkers

In this paper, we study 𝑛-dimensional complete self-shrinkers in R n + 1 Rⁿ⁺¹ with constant squared norm 𝑆 of the second fundamental form. We partially resolve the conjecture on 𝑛-dimensional complete self-shrinkers in R n + 1 Rⁿ⁺¹ with constant squared norm 𝑆 of the second fundamental form. Furthermore, if the scalar curvature of an 𝑛-dimensional self-shrinker is constant, then we prove that the scalar curvature 𝑅 satisfies R ≤ n − 1 R≤ n-1 . We also classify 𝑛-dimensional complete self-shrinkers in R n + 1 Rⁿ⁺¹ with non-negative constant scalar curvature.