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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)

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
##### Keywords:
first eigenvalue; Laplace-Beltrami operator
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