Stable bundles and the first eigenvalue of the Laplacian

In this paper we study the first eigenvalue of the Laplacian on a compact Kaehler manifold using stable bundles and balanced bases.

In this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let M be a compact Kähler manifold of complex dimension n and E a holomorphic vector bundle of rank r over M. If E is globally generated and its Gieseker point T e is stable, then for any Kähler metric g on M λ 1 (M,g) ≤ 4π h 0 (E)/r(h 0 (E) - r ̇ 〈 {C 1 (E) ∪ [ω] n 1,[M] 〉 (n - 1)!vol(M,[ω] where ω = ω g is the Kähler form associated to g. By this method we obtain, for example, a sharp upper bound for λ 1 of Kähler metrics on complex Grassmannians. © 2007 Mathematica Josephina, Inc.