Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces

Xin Nie

Published 2022-06-27Version 1

We associate to each cyclic $\mathrm{SO}_0(n,n+1)$-Higg bundle a special type of minimal surface in the pseudo-hyperbolic space $\mathbb{H}^{n,n}$ (if $n$ is even) or $\mathbb{H}^{n+1,n-1}$ (if $n$ is odd) with a notion of Gauss map that recovers the equivariant minimal mapping into the symmetric space. By showing that such minimal surfaces are infinitesimally rigid, we get a new proof, for $\mathrm{SO}_0(n,n+1)$, of Labourie's theorem that the Hitchin map restricts to an immersion on the cyclic locus, and extend it to Collier's components. This implies Labourie's former conjecture in the case of the exceptional group $G_2'$, for which we also show that the minimal surfaces are $\boldsymbol{J}$-holomorphic curves of a particular type in the almost complex $\mathbb{H}^{4,2}$.