Meromorphic differentials with twisted coefficients on compact Riemann surfaces

We apply the Dirichlet’s principle to a modified energy functional on Riemann surfaces to reprove the existence of harmonic metrics with certain prescribed singularities due to Simpson, Sabbah and Biquard–Boalch, and hence of differentials with twisted coefficients of the second and third kinds. As a by-product, this generalizes the classical theory of Abelian differentials on a compact Riemann surface to the case of twisted coefficients. This also proposes a more natural approach for general existence of harmonic metrics in the higher dimensional case. The author supported partially by NSF of China (No. 10471105, 10771160).