Energy of Harmonic Maps and Gardiner's Formula

It is shown that the usual first variational formula for the energy of a harmonic map (or equivariant harmonic map) on a two dimensional domain with respect to the conformal structure extends to case of nonpositively curved metric space targets. As applications, we recover Gardiner's formula for the variation of the Hubbard-Masur differential and a proof of the existence and uniqueness of quadratic differentials realizing a pair of measured foliations that fill a surface.