Approximate Hermitian-Einstein connections on principal bundles over a compact Riemann surface

Let X be a compact connected Riemann surface and G a connected reductive complex affine algebraic group. Given a holomorphic principal G-bundle E_G over X, we construct a C^∞ Hermitian structure on E_G together with a 1-parameter family of C_∞ automorphisms {F_t}_t∈ℝ of the principal G-bundle E_G with the following property: Let ∇^t be the connection on E_G corresponding to the Hermitian structure and the new holomorphic structure on E_G constructed using F_t from the original holomorphic structure. As t → -∞, the connection ∇^t converges in C^∞ Fréchet topology to the connection on E_G given by the Hermitian-Einstein connection on the polystable principal bundle associated to E_G. In particular, as t → -∞, the curvature of ∇^t converges in C^∞ Fréchet topology to the curvature of the connection on E_G given by the Hermitian-Einstein connection on the polystable principal bundle associated to E_G. The family {F_t}_t∈ℝ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle E on X, in [6] a 1-parameter family of C^∞ automorphisms of E is constructed such that as t → -∞, the curvature converges, in C⁰ topology, to the curvature of the Hermitian-Einstein connection of the associated graded bundle.