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 Te 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 = ωg 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.
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Arezzo, C., Ghigi, A. & Loi, A. Stable bundles and the first eigenvalue of the Laplacian. J Geom Anal 17, 375–386 (2007). https://doi.org/10.1007/BF02922088
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DOI: https://doi.org/10.1007/BF02922088