Scalar curvature bounds and rigidity of Ricci shrinkers

We prove that a complete noncompact gradient shrinking Ricci soliton of dimension at least two is a finite quotient of an Einstein cylinder if its scalar curvature is bounded below by the cylindrical value. The corresponding pointwise strict bound implies compactness. We also establish one-endedness criteria using curvature information at critical points of the potential, including degenerate ones, and obtain sharp one-endedness and rigidity results under global Ricci norm bounds.