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Stable bundles and the first eigenvalue of the Laplacian. (English) Zbl 1128.58013
From the authors’ abstract: 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 $$M$$ $\lambda_{1}(M,g)\leq\frac{4\pi h^{0}(E)}{r(h^{0}(E)-r)}\cdot \frac{\langle c_{1}(E)\cup[\omega]^{n-1},[M]\rangle}{(n-1)!\text{vol}(M,[\omega])},$ where $$\omega=\omega_{g}$$ is the Kähler metric associated to $$g$$. By this method we obtain, for example, a sharp upper bound for $$\lambda_ {1}$$ of Kähler metrics on complex Grassmannians.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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