New Sphere Theorems under Curvature Operator of the Second Kind

We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\barλ, \end{equation*} where ${\lambda }_1\le \cdots \le {\lambda }_{(n-1)(n+2)\mathrm{/}2}$ are the eigenvalues of $\mathring{R}$, $\bar{\lambda }$ is their average, and $\theta >-1$. Under such conditions with optimal $\theta$ depending on $n$ and $\alpha$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to $\theta =0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.