Variational Problem of Some Quadratic Functionals in Complex Analysis

Suppose we are given a pair of marked closed Riemann surfaces S, T with the same genus. We minimize the modified Dirichlet integral for mappings of S onto T, which is endowed with some weight in terms of conformal metric on the target surface T. In this paper, it is demonstrated that the minimal harmonic map exists, and it proves to be Teichm¨uller