Infinitesimal Rigidity of Cyclic Surfaces for SL(n,ℂ)

Labourie introduced cyclic surfaces in 2014 to study mapping class group–equivariant parameterizations of higher Teichmüller spaces, particularly Hitchin components for split real rank-two groups. A key step in his approach is the proof of infinitesimal rigidity of cyclic surfaces. This rigidity has since been extended to settings such as maximal SO(2,n)-components, cyclic G_2'-Higgs bundles, cyclic SO(n,n+1)-Higgs bundles and cyclic SL(2n+1,ℝ)-Higgs bundles, using either variations of Labourie‘s method or more geometric arguments.

These results, however, apply only to certain real forms of Lie groups. In this talk, we study cyclic Higgs bundles defined by an order-m automorphism of the complex group SL(n, ℂ) and establish infinitesimal rigidity for the associated cyclic surfaces. This framework recovers all previous rigidity results in a unified way.

This is joint work with Qiongling Li.