New Sphere Theorems under Curvature Operator of the Second Kind

TL;DR

We study lower bounds on the curvature operator of the second kind via the two-parameter cone $\mathcal{C}(\alpha,\theta)$, with $\theta>-1$, generalizing prior $\theta=0$ results. The main contributions are two differentiable sphere theorems in dimensions $3$ and $4$ under the optimal threshold $\theta=\bar{\Theta}_{n,\alpha}$, a higher-dimensional homological sphere theorem, and a curvature characterization of Kähler space forms, extending the scope of the curvature-cone framework $\mathcal{C}(\alpha,\theta)$. Sharpness is demonstrated using model spaces such as $\mathbb{S}^{n-1}\times\mathbb{S}^1$, $\mathbb{CP}^m$, and products, with connections to Ricci and isotropic curvature via Bochner-type arguments and a weight principle. The results open avenues for further refinement in higher dimensions and raise questions about Ricci-flow preservation of the cone conditions.

Abstract

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 $λ_1 \leq \cdots \leq λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\barλ$ 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 Kähler 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.

Theorems & Definitions (42)

• Theorem 1.1: CGT23, Li21, and NPW22
• Definition 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1