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Let $M$ be a closed Riemannian manifold and $\Delta$ its Laplace-Beltrami operator. Then we have Yau's estimate on the first (non-zero) eigenvalue $\lambda_1>0$ of $\Delta$ acting on functions in terms of bounds on the Ricci curvature, the diameter and the volume of $M$.

1. Is there any estimate like this for $\Delta$ acting on forms?

2. Is an estimate like this know to be false assuming only Ricci, volume and diameter bounds?

3. If we assume a "nice" degeneration of Riemannian manifolds, like K3 surfaces close arising from the Kummer construction degenerating to $T^4/{\bf Z}_2$, can we control first eigenvalue of $\Delta$ on forms (from below) along a degenerating sequence?

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Closely connected to what you are talking about is Reilly's Formula (it deals with manifolds with boundary, but estimates on $\Delta$ on manifolds without boundary are possible using it). Here is a paper extending this to forms: http://arxiv.org/abs/1003.0817. I'm sure there are bound to be interesting estimates in there, or allusions as to where else to look. Beware that the curvature terms you have to control aren't as pretty as just Ricci curvature, unfortunately.

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  • $\begingroup$ I should have said 'estimates on $\lambda_1$'. $\endgroup$ – Michael Coffey Apr 16 '11 at 9:39

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