Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality

This paper establishes a conformally invariant $L^2$ surjectivity criterion for real-linear Cauchy--Riemann operators, in terms of the contracted Chern curvature and the $L^2$-energy of the zero-order term. We show that the supremum of the integrated curvature term over all Hermitian metrics is determined exactly by the minimal Harder--Narasimhan slope, leading to an asymptotic Harder--Narasimhan criterion. A filtration principle is developed that reduces surjectivity to successive quotient operators. These results yield higher-rank automatic transversality criteria for pseudoholomorphic curves, with applications to the standard nearly Kähler $6$-sphere.