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Riemannian metrics with large \(\lambda_ 1\). (English) Zbl 0820.58056
On a closed Riemannian manifold \((M,g)\) let \(\lambda_ 1(g)\) denote the smallest positive eigenvalue of the Laplace-Beltrami operator. For surfaces \(S\) it is known (Hersch and Yang/Yau) \[ \lambda_ 1(g) \cdot \text{vol}(S,g) \leq 8\pi \cdot (\text{genus}(S) + 1). \] In dimension \(n \geq 3\) the situation is different as is shown by
Theorem. Every closed manifold \(M\) of dimension \(n \geq 3\) admits a Riemannian metric \(g\) of volume 1 with arbitrarily large \(\lambda_ 1(g)\).
The authors sketch their proof of this theorem. The idea is to start with \(S^ n\) where such metrics are known to exist and then glue in the manifold \(M\) in a metrically very small ball.
Reviewer: C.Bär (Freiburg)

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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