A New Parabolic Flow in Kähler Manifolds

Abstract

This is a follow-up work of my earlier paper [8]. In [8], we study the lower bound of the K energy on the Kahler manifold when the first Chern class is negative. This is an important problem in Kahler geometry since the existence of lower bound of the K energy is the pre-condition for the existence of constant scalar curvature metric problem (cf. [2] and [9]). According to a decomposition formula in [8]2, the problem is reduced to the problem of solving the existence of critical metrics of a new functional J introduced both in our paper [8] and Donaldson’s work [11]. For convenience, we include its definition below. The existence problem is completed solved in Kahler surface. However, the existence problem in general dimension is still open. In this paper, we try to understand the general existence problem in Kahler manifold via flow method. Let (V n, ω0) be a Kahler manifold and ω0 be any Kahler form in V. Consider the space of Kahler potentials