arXiv subjects

Zhi-Lin Dai

Publications and source records attributed to Zhi-Lin Dai.

2 recordsLinked to original sources

arXiv2026

K{\"a}hler Einstein manifolds and the Calabi curvature operator

In this paper, we study the Calabi curvature operator on K{\"a}hler manifolds. First, we prove that if the Calabi curvature operator on K{\"a}hler manifolds satisfies $\frac{{n\left(n + 1\right)}}{2}$-positive (nonnegative), $\frac{{n + 1}}{2}$-positive (nonnegative), and $\left( n-1 \right)$-positive (nonnegative), then the scalar curvature, Ricci curvature, and orthogonal Ricci curvature are positive (nonnegative), respectively. Second, we show that any compact K{\"a}hler Einstein manifold satisfying the condition $$\lambda_1+\dots+\lambda_{\alpha}\ge -{\alpha}\theta(n,{\alpha})\bar\lambda,\; {\alpha}\le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.

math.DG↗

arXiv2023

Einstein manifolds and curvature operator of the second kind

We prove that a compact Einstein manifold of dimension $n\geq 4$ with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension $n\geq 11$ with $\left [ \frac{n+2}{4} \right ]$-nonnegative curvature operator of the second kind, $4\ (\mbox{resp.},8,9,10)$-dimensional compact Einstein manifolds with $2$-nonnegative curvature of the second kind and $5$-dimensional compact Einstein manifolds with $3$-nonnegative curvature of the second kind are constant curvature spaces. Combing with Li's result [10], we have that a compact Einstein manifold of dimension $n\geq 4$ with $\max\{4,\left [ \frac{n+2}{4} \right ]\}$-nonnegative curvature operator of the second kind is a constant curvature space.

math.DG↗