Holomorphic $mathfrak{sl}(2,mathbb C)$-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

For every integer $g ,geq, 2$ we show the existence of a compact Riemann surface $Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${mathcal O}^{oplus 2}_{Sigma}$ admits holomorphic connections with $text{SL}(2,{mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Sigma$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.