Asymptotics of Higgs bundles in the Hitchin component

In this paper we pursue a more geometric approach to compactification of the Hitchin component. Our main motivation is Wolf's harmonic map interpretation of Thurston's compactification of Teichmüller space with measured foliations. Using Hitchin's parameterization of the Hitchin component by holomorphic differentials, we study asymptotics of certain rays of representations. More precisely, along these rays we solve the Hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection. The asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n,R) proves a conjecture of Katzarkov, Noll, Pandit and Simpson [17] on the Hitchin WKB problem in our setting.