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We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darbouxcoordinates on the moduli space $\mathcal{M}$ of weakly parabolic$SL(2,\mathbb{C})$-Higgs bundles. For generic Higgs bundles($\mathcal{E},R\Phi)$ with $R\gg 0$ the coordinates are shown to be dominatedby a leading term that is given by the coordinates for a corresponding simplerspace of limiting configurations and we prove that the deviation from thelimiting term is given by a remainder that is exponentially suppressed in $R$.  We then use this result to solve an associated Riemann-Hilbert problem andconstruct a twistorial hyperk\"ahler metric $g_{\text{twist}}$ on$\mathcal{M}$. Comparing this metric to the simpler semiflat metric$g_{\text{sf}}$, we show that their difference is$g_{\text{twist}}-g_{\text{sf}}=O\left(e^{-\mu R}\right)$, where $\mu$ is aminimal period of the determinant of the Higgs field.

Hyperk\"ahler metrics on the moduli space of weakly parabolic Higgs bundles