Unstable minimal surfaces in ${\mathbb{R}}^n$ and in products of hyperbolic surfaces

Abstract

We prove that every unstable equivariant minimal surface in ${\mathbb{R}}^n$ produces a maximal representation of a surface group into ${\prod }_{i=1}^n\mathrm{PSL}(2,\mathbb{R})$ together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in ${\mathbb{R}}^n$ to a surface in a product of $\mathbb{R}$-trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.