Convergence of Yang-Mills-Higgs flow for twist Higgs pairs on Riemann surfaces

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle (E,H) over Riemann surface X. It is already known the gradient flow with initial data (A ₀, ? ₀) converges to a critical point (A _∞, ? _∞). Using a modified Chern-Weil type inequality, we prove that the limiting twist Higgs bundle (E, \(d''_{A_\infty }\) ? _∞) coincides with the graded twist Higgs bundle defined by the Harder-Narasimhan-Seshadri filtration of the initial twist Higgs bundle (E, \(d''_{A_0 }\) , ? ₀), generalizing Wilkin’s results for untwist Higgs bundle.