On Aubin-Lichnerowicz's estimate and domination inequality. (Q1428218)

Let \((M,g)\) be a compact \(n\)-dimensional manifold with a convex boundary. Assume the Ricci curvature is bounded below by \(R\). Then Lichnerowicz estimated that \(\lambda_1(M)\geq {n\over{n-1}}R\) for the least positive eigenvalue \(\lambda_1\) of the Hodge-Kodaira Laplacian. Subsequently \textit{T. Aubin} [J. Funct. Anal. 57, 143--153 (1984; Zbl 0538.53063)] showed \(\lambda_1(M)\geq2R\) if \((M,g)\) is Kähler. In the present note, the author extends this estimate to the category of line bundles. Theorem: Let \((E,h)\) be a Hermitian line bundle with Ricci form \(F_{i\bar j}\) over a compact Kähler manifold \((M,g)\) with Ricci form \(R_{i\bar j}\). Suppose that \(R_{i\bar j}+F_{i\bar j}\geq R\). Then the least positive eigenvalue \(\lambda_1\) of the Hodge-Kodaira Laplacian satisfies \(\lambda_1\geq R\).