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Upper bound for the first eigenvalue of algebraic submanifolds. (English) Zbl 0814.53040
Let $$0 < \lambda_ 1$$ be the first nonzero eigenvalue of the Laplacian on a compact Riemann manifold $$M$$. In 1970, Hersch gave a sharp upper bound for $$\lambda_ 1$$ in terms of the volume alone if $$M$$ is the 2- sphere with any metric; Berger has pointed out that Hersch’s theorem fails for higher dimensional spheres. Generalizations of Hersch’s theorem have been provided by Yang and Yau for other compact oriented surfaces. Let $$\sigma$$ be the Kähler form on the complex projective space $$CP^ N$$ of dimension $$N$$. In this paper, the authors study the complex category and show:
Theorem: Let $$M^ m$$ be an $$m$$-dimensional compact complex manifold admitting a holomorphic immersion $$\Phi$$ into $$CP^ N$$. Suppose that $$\Phi (M)$$ is not contained in any hyperplane of $$CP^ N$$. Then for any Kähler metric on $$M$$, $\lambda_ 1 (M,\omega) \leq 4m (N+1)d \bigl( [\Phi], [\omega] \bigr)/N$ where the holomorphic immersion degree (a homological invariant) is given by: $d \bigl( [\Phi], [\omega] \bigr) : = \Bigl\{ \int_ M \Phi^* (\sigma) \wedge \omega^{m-1} \Bigr\} \cdot \Bigl\{ \int_ M \omega^ m \Bigr\}^{-1}.$
Reviewer: P.Gilkey (Eugene)

##### MSC:
 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
Laplacian; holomorphic immersion; Kähler metric
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