A new flow solving the LYZ equation in Kähler geometry

Abstract

We introduce a new flow to the LYZ equation on a compact Kähler 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 $\href{https://dx.doi.org/10.4310/CJM.2020.v8.n2.a4}{[5]}$ by the continuity method. Moreover, as an application of the flow, we show that on a compact Kähler 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 Kähler metric.