A construction of hyperkähler metrics through Riemann–Hilbert problems II

$\def\Link{\href{ https://doi.org/10.48550/arXiv.1701.08188}{[3]}}$We develop the theory of Riemann–Hilbert problems necessary for the results in $\Link$. In particular, we obtain solutions for a family of non-linear Riemann–Hilbert problems through classical contraction principles and saddle-point estimates. We use compactness arguments to obtain the required smoothness property on solutions. We also consider limit cases of these Riemann–Hilbert problems where the jump function develops discontinuities of the first kind together with zeroes of a specific order at isolated points in the contour. Solutions through Cauchy integrals are still possible and they have at worst a branch singularity at points where the jump function is discontinuous and a zero for points where the jump vanishes.

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Published 5 January 2024