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 $\lambda_1(\mathbb{S}^{n}) = n$)?