Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the (positive) Laplace-Beltrami operator $\Delta$ satisfies
\begin{equation}
\lambda_1(M) \geq \frac{\pi^2}{d^2},  \tag{1}
\end{equation}
where $d$ is the diameter of $M$. Another well known result is the Lichnerowicz theorem: if $\mathrm{Ric} \geq (n-1)K >0$ then
$$\lambda_1 \geq n K.$$
All these lower bounds require some assumption on the curvature. 


*

*Are there universal (i.e. curvature-independent) lower bounds for $\lambda_1$ on a closed Riemannian manifold? 

*More precisely, does the inequality (1) hold even with no assumption on the Ricci curvature? 

*

*If not, are there counter-examples?

*If yes, is this the best bound one can achieve in this sense (clearly (1) is not sharp as  $\lambda_1(\mathbb{S}^{n}) = n$)?


 A: The answer is no.
To see this consider two (2D) sphere's of constant curvature 1 and a (flat) cylinder of unit length and radius $\epsilon$.  Cut out a disk from each sphere and glue in the cylinder (and smooth everything out in a small neighborhood of the gluing).  By Gauss Bonnet this must introduce negative curvature.
Now take a function which is identically 1 on one sphere and identically -1 on the other sphere and which linearly interpolates between the two on the cylinder.  It's clear that this can be chosen to be orthogonal to the constants.  Moreover, this function has $L^2$ norm which is (up to a small error) independent of $\epsilon$ (and positive) while the Dirichlet energy is on the order of $\epsilon$.  In particular, the variational characterization of $\lambda_1$ implies that $\lambda_1$ is smaller than $C\epsilon$ for some fixed constant $C$ -- i.e. as small as one likes. 
A: A well-known result of Y. Colin de Verdière states that given any compact connected manifold $M$, with $\dim M\geq 3$, and any finite sequence $0<a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0< a_1\leq\dots\leq a_k$. The original paper can be found here. Therefore, there is no hope for a "universal" lower bound on $\lambda_1(M)$.
