Chern class inequalities implied by partial differential equations

Determine whether a partial differential equation implies a Chern class inequality.

Background

The paper studies the Kähler–Yang–Mills equations, which couple a scalar-curvature equation for a Kähler metric with a Hermitian–Einstein equation for a holomorphic vector bundle. In discussing the role of the Kobayashi–Lübke inequality in the convexity argument for the alpha-K-energy, the authors raise the broader unresolved issue of whether differential equations can force inequalities involving Chern classes.

This question is included because such an implication could provide a systematic way to derive topological constraints from coupled geometric PDEs and might guide the formulation of appropriate couplings and stability conditions.