Limiting configurations for the SU(1,2) Hitchin equation

Abstract

We study the limiting behavior of the solutions $h_t$ of the Hitchin's equation associated with a family of stable SU(1,2) Higgs bundles $(L,F,t\beta,t\gamma)$ on a compact connected Riemann surface $X$ as $t\to\infty$ under the assumption that the quadratic differential $q=\beta\cdot\gamma$ have simple zeros at $D$. The spectral data of the SU(1,2) Higgs bundle $(L,F,\beta,\gamma)$ can be represented by a Hecke modification of $V=L^{-2}K_X\oplus LK_X$. We show by a gluing construction that after appropriate rescaling, the limit is given by a metric on $V$ singular at $D$, induced by harmonic metrics adapted to parabolic structures on $L$ and on $K_X$ at $D$. We give rules to determine the parabolic weights of the limit.