Complex projective structures over Riemann surfaces and solutions to Hitchin's equations

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd [3]

R. Gunning in 1967 defined a projective coordinate system of a compact Riemann surface as one for which the transition functions are given by Möbius transformations. Such structures are alternatively described by particular flat bundles called by Gunning at the time "indigenous bundles". In modern terminology, this pertains to the structure of an $\text{SL} \left(n, \mathbb{C}\right)$-oper as introduced by A. Beilinson and V. Drinfeld, who, more generally, introduced $G$-opers for any simple simply connected complex Lie group $G$. We shall focus in this talk on the relationship of these structures to certain families of solutions to Hitchin's equations. Joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.