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 $0\leq a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0\leq a_1\leq\dots\leq a_k$. The original paper can be found [here][1]. Therefore, there is no hope for a "universal" lower bound on $\lambda_1(M)$.


  [1]: https://www-fourier.ujf-grenoble.fr/~ycolver/All-Articles/87a.pdf