A new flow solving the LYZ equation in K\"ahler geometry

Abstract

We introduced a new flow to the LYZ equation on a compact K\"ahler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime solution converges to the solution of the LYZ equation, which was solved by Collins-Jacob-Yau [5] by the continuity method. Moreover, as an application of the flow, we show that on a compact K\"ahler surface, if there exists a semi-subsolution of the LYZ equation, then our flow converges smoothly to a singular solution to the LYZ equation away from a finite number of curves of negative self-intersection. Such a solution can be viewed as a boundary point of the moduli space of the LYZ solutions for a given K\"ahler metric.