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The space of volume forms

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arxiv 0810.3880 v2 pith:B5E3OT7S submitted 2008-10-21 math.DG math.AP

Xiuxiong Chen , Weiyong He This is my paper
classification math.DGmath.AP
keywords spacevolumeequationformsmetricgeodesicalexandrovcompact
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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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