Biharmonic Hermitian vector bundles over compact Kähler manifolds and compact Einstein Riemannian manifolds

Abstract

We show, for every Hermitian vector bundle $\pi:\,(E,g)\rightarrow (M,h)$ over a compact Kähler Einstein manifold $(M,h)$, if the projection $\pi$ is biharmonic, then it is harmonic. On a biharmonic Hermitian vector bundle over a compact Riemannian manifold with positive Ricci curvature, we show a new estimate of the first eigenvalue of the Laplacian.