For a cyclic Higgs bundle over a noncompact Riemann surface, we study the space of Toda-like harmonic metrics depending on the asymptotic behavior of the deteminant of the Higgs field. We will attach the problem in both the Hitchin-Kobayashi method and the partial differential equation method. This is a joint work with Takuro Mochizuki.

I will review the basic concepts of Higgs bundles and the moduli space of Higgs bundles, and then focus the non-Abelian Hodge correspondence: relation between the moduli space of Higgs bundles with harmonic maps and the character variety. I will also explain some examples: rank two Higgs bundles, cyclic Higgs bundles, variation of Hodge structure and so on.

Hitchin component for $SL(n,R)$ is the component in the space of surface group representations into $SL(n,R)$ which can deform to Fuchsian locus. The Hitchin component is in correspondence with the moduli space of $SL(n,R)$-Higgs bundles. I will introduce recent work with Brian Collier on asymptotic behaviors of families in Hitchin component in terms of certain families of Higgs bundles. Namely, given a family of Higgs bundles by scaling Higgs field by $t$, we analyze the asymptotic behavior of the corresponding representations as $t$ goes to $\infty$ in two special cases.