Using Hitchin's parameterization of the Hitchin-Teichm\"uller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.

Original language

Undefined/Unknown

Journal

Advances in Mathematics

Volume

307

State

E-pub ahead of print - Feb 2017

Bibliographical note

48 pages, v2: ~20 pages added to v1, substantial changes were made to the proof of parallel transport asymptotics