Table of Contents
Geometry
Volume 2014, Article ID 968064, 6 pages
http://dx.doi.org/10.1155/2014/968064
Research Article

Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA

Received 14 June 2013; Accepted 11 October 2013; Published 2 January 2014

Academic Editor: Reza Saadati

Copyright © 2014 Daniel Guan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.