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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)

53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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