The space of volume forms
Differential Geometry
2010-04-16 v2 Analysis of PDEs
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.
Cite
@article{arxiv.0810.3880,
title = {The space of volume forms},
author = {Xiuxiong Chen and Weiyong He},
journal= {arXiv preprint arXiv:0810.3880},
year = {2010}
}
Comments
A gap in Section 5 is fixed; references added.