2009/03/13 by La Nave, Gabriele, Tian, Gang
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.0903.2413
In this paper, we first show an interpretation of the Kähler-Ricci flow on a manifold X as an exact elliptic equation of Einstein type on a manifold M of which X is one of the (Kähler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on X will do). Such an equation is called V-soliton equation, which can be regarded as a generalization of Kähler-Einstein equations or Kähler-Ricci solitons. As in the case of Kähler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampere type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kähler manifold M. One of our motivations is to apply the interpretation to studying finite time singularities of the Kähler-Ricci flow.