The space of volume forms

Details The space of volume forms The space of volume forms

2008-10-21

arxiv.org/abs/0810.3880v2

Mathematics

Differential Geometry

A gap in Section 5 is fixed; references added.

Scientific paper

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.

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