Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces

Abstract

We introduce a type of minimal surface in the pseudo-hyperbolic space Hn,n (with n even) or Hn+1,n−1 (with n odd) coming from cyclic SO0(n,n+1)-Higg bundles. By showing that these surfaces are saddle-type critical points for the area functional, and hence infinitesimally rigid, we get a new proof, for SO0(n,n+1), of Labourie's theorem that the holonomy map restricts to an immersion on the cyclic locus of Hitchin base, and extend it to Collier's components. This implies Labourie's former conjecture in the case of the exceptional group G2′, for which we also show that these minimal surfaces are J-holomorphic curves of a particular type in the almost complex H4,2.