In this talk we will discuss some aspects of the problem of finding critical points of $\lambda_1$ and its invariant cousin restricted to a special set of metrics, to be specific, the set of toric Kähler metrics on a toric manifold.

We will start by reviewing some classical results on bounding and extremizing $\lambda_1$ and then we proceed to describe the setting in which we want to work, namely, that of toric Kähler manifolds. In this setting we discuss some recent results on boundeness of $\lambda_1$ and non-existence of critical points for the invariant $\lambda_1^\mathbb{T}.$