I will describe two interesting and closely related moduli spaces: the nilpotent cone in the moduli spaces of Higgs bundles for $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{PSL}_2(\mathbb C)$, and the moduli space of equivariant minimal surfaces in the hyperbolic 3-space.

A deep understanding of these objects is important because of their relations with several fundamental constructions in geometry: singular fibers of the Hitchin fibration, branes, mirror symmetry, branched hyperbolic structures, minimal surfaces in hyperbolic 3-manifolds and so on.

A stratification of the nilpotent cone is well known and was rediscovered by many people. The closures of the strata are the irreducible components of the nilpotent cone. The talk will focus on describing the intersections between the different irreducible components.

This is joint work with Qiongling Li and Andrew Sanders.