第二类曲率算子的两个特征值条件下爱因斯坦流形的刚性
Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
- Soochow University（苏州大学）
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AI总结：
本文证明了在第二类曲率算子两个特征值下界条件下，闭爱因斯坦流形（维数n≥6）的刚性定理，得出流形为平坦或球空间形式。
AI中文摘要：
我们证明了在第二类曲率算子的两个下界条件下，维数$n\ge6$的闭爱因斯坦流形的刚性定理。更精确地，对于$L_1, L_2\ge0$以及某个在适当范围内的实数$\alpha>1$，我们考虑\\[ \lambda_1\ge-L_1\bar\lambda, \qquad \frac1\alpha\sum_{j=1}^{\alpha}\lambda_j \ge-L_2\bar\lambda. \\] 这里$\lambda_1\le\cdots\le\lambda_N$是第二类曲率算子$\mathring R$的特征值，$N=(n-1)(n+2)/2$，且$\bar\lambda=N^{-1}\sum_{j=1}^N\lambda_j$。设$\theta(n, \alpha)$为\cite[Theorem~1.1]{CW26}中出现的常数；见(2)。在本文考虑的参数范围内，$L_2>\theta(n, \alpha)$，因此第二个条件严格弱于\cite{CW26}中的相应条件，而第一个条件与该条件互不蕴含。在$L_1$和$L_2$之间满足一个额外的显式关系时，我们证明该流形要么是平坦的，要么是球空间形式。
英文摘要：
We prove a rigidity theorem for closed Einstein manifolds of dimension $n\ge6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1, L_2\ge0$ and some real number $α>1$ in a suitable range, we consider \[ λ_1\ge-L_1\barλ, \qquad \frac1α\sum_{j=1}^αλ_j \ge-L_2\barλ. \] Here $λ_1\le\cdots\leλ_N$ are the eigenvalues of the curvature operator of the second kind $\mathring R$, $N=(n-1)(n+2)/2$, and $\barλ=N^{-1}\sum_{j=1}^Nλ_j$. Let $θ(n, α)$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, $L_2>θ(n, α)$, so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between $L_1$ and $L_2$, we prove that the manifold is either flat or a spherical space form.
