TITLE: Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold
QUESTION [11 upvotes]: 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  $\lambda_1(\mathbb{S}^{n}) = n$)?

REPLY [5 votes]: 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 $\lambda_1(M)$.