First eigenvalue estimates on complete Kähler manifolds

Abstract

Let $(M,\ddot{I}{‰}_g)$ be a complete Kähler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC}\ge 2$, then the first eigenvalue $\hat{I}{»}_1$ of the Laplacian on $(M,\ddot{I}{‰}_g)$ satisfies $$\hat{I}{»}_1\ge \frac{320(n-1)+576}{81(n-1)+144}.$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.