New Sphere Theorems under Curvature Operator of the Second Kind

We investigate Riemannian manifolds (Mⁿ,g) whose curvature operator of the second kind R satisfies the condition {equation*} α⁻¹ (λ_1 +⋯ +λα) > - θ λ̄, {equation*} where λ₁ ≤ ⋯ ≤ λ(n-1)(n+2)/2 are the eigenvalues of R, λ̄ is their average, and θ > -1. Under such conditions with optimal θ depending on n and α, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of K\"ahler space forms. These results generalize recent works corresponding to θ =0 of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.