The space of volume forms

Donaldson introduced an interesting geometric structure (the Donaldson metric) on the space of volume forms for any compact Riemannian manifold, which has nonpositive sectional curvature formally. The geodesic equation and its perturbed equations are fully nonlinear elliptic equations. These equations are also equivalent to two free boundary problems of the Laplacian equation, and it also has a relationship with many interesting problems, such as Nahm's equation. In this paper, we solve these equations and demonstrate the geometric structure of the space of volume forms; in particular, we show that the space of volume forms with the Donaldson metric is a metric space with non-positive curvature in Alexanderov sense.