Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let $M$ be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on $M$ to be extremal for $\lambda_1$ with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice $\Gamma$ of $\mathbb{R}^n$, the flat metric $g_{\Gamma}$ induced on $\mathbb{R}^n/\Gamma$ from the standard metric of $\mathbb{R}^n$ is extremal (in the previous sense). In the second part, we give, for any $\Gamma$, an upper bound of $\lambda_1$ on the conformal class of $g_{\Gamma}$ and exhibit a class of lattices $\Gamma$ for which the metric $g_{\Gamma}$ maximizes $\lambda_1$ on its conformal class.