Let $ (M,g)$ be a connected, oriented, compact Riemann surface with positive constant curvature $ K_g$ , and let $ P\to M$ be a principal $ S^1$ -bundle on $ M$ . Can we find a Yang-Mills connection $ A$ on $ P$ such that the norm of the curvature $ 2$ -form $ F_A\in A^2(M,\mathbb R)$ is equal to $ K_g$ (i.e. $ |F_A|_g=K_g$ )?
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