Stable bundles and the first eigenvalue of the Laplacian

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\(\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup \omega ]^{n - 1} ,M]} \right\rangle }}{{(n - 1)!vol(M,\omega ])}}\) where ω = ω_g is the Kähler form associated to g.By this method we obtain, for example, a sharp upper bound for λ₁ of Kähler metrics on complex Grassmannians.