Toda equations and cyclic Higgs bundles over non-compact surfaces

For Higgs bundles over compact Kahler manifods, it is known by Hitchin and Simpson that the existence of Hermitian-Einstein metric is equivalent to the polystability of the Higgs bundle. There are some generalizations to non-compact cases. On a Riemann surface with a holomorphic r-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a Hermitian-Einstein metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an r-differential which is not necessarily meromorphic at infinity. In particular, we discuss the Hermitian-Einstein metrics on the cyclic Higgs bundles determined by r-differentials. This is joint work with Takuro Mochizuki (Kyoto University).