General construction of the g-function beyond the genus-0 ansatz

Develop an efficient general method to construct the Deift–Zhou g-function for the 3×3 Riemann–Hilbert problem associated with the (3,4) string equation on three-sheeted Riemann surfaces of arbitrary genus and verify the required lens-opening inequalities, thereby removing the current restriction to genus-0 spectral curves.

Background

For the steepest-descent analysis, the authors construct a g-function on a three-sheeted Riemann surface to normalize oscillatory factors. They restrict to the case where the underlying Riemann surface has genus 0, which allows explicit rational parametrizations and verification of lensing inequalities.

A general approach applicable to higher-genus three-sheeted curves is not provided, limiting the scope of the asymptotic analysis and indicating a methodological gap.