The space of volume forms

Abstract

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its perturbed equation and prove that the space of volume forms is an infinite dimensional non-positively curved metric space in the sense of Alexandrov.

A gap in Section 5 is fixed; references added.

References in corpus (3)

• The Space of Kähler metrics (II)
• Space of Kähler metrics III--On the lower bound of the Calabi energy and geodesic distance
• The Donaldson equation

Cited by in corpus (5)

• The Donaldson equation
• Regularity of the geodesic equation in the space of Sasakian metrics
• Partial Legendre transforms of non-linear equations
• Entire solutions of Donaldson's equation
• A maximum rank problem for degenerate elliptic fully nonlinear equations