On complete Kähler manifolds with HSC ≥ 2, the first eigenvalue of the Laplacian is at least (320n+256)/(81n+63), which tends to 320/81 as n grows.
The rigidity of eigenvalues on K\"ahler manifolds with positive Ricci lower bound
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First eigenvalue estimates on complete K\"ahler manifolds
On complete Kähler manifolds with HSC ≥ 2, the first eigenvalue of the Laplacian is at least (320n+256)/(81n+63), which tends to 320/81 as n grows.