Moser Vector Fields and Geometry of the Mabuchi Moduli Space of K？hler Metrics

There is a natural Moser type transformation along any curve in the moduli spaces of K？hler 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 K？hler metrics. This is crucial in the proof of the equivalence between the existence of the K？hler 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 K？hler-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. 1. Introduction In the study of existence of K？hler-Einstein metrics Mabuchi introduced the geodesic equations in [1]. It turns out that the special homogeneous complex Monge-Ampére equation Semmes considered in [2] is exactly the same equation considered by Mabuchi. Let be a compact K？hler manifold, and let be a real smooth function with variables of time and point . Regarding as the real part of a complex number , is a function of and . If satisfies the complex homogeneous Monge-Ampére equation and is a K？hler metric for , then is a geodesic in the Mabuchi moduli space of the K？hler metrics. Locally if is the K？hler potential of and , by letting we see that the equation is the same as . It was observed that the kernel of induced a vector field of a certain Moser type transformation. And the Moser type vector field can be applied to any general curve in the space of the K？hler metrics. In this paper we will apply this Moser type transformation to give an explicit construction for the parallel transformation along any curve in the Mabuchi moduli spaces of K？hler metrics. The existence of the parallel transformation was proven in [1, page 234] by integration of a vector field and then in [3], without knowing Mabuchi’s first proof, obtained by solving one of the two quasilinear equations which led to the existence of the