Coupled continuity equations for constant scalar curvature Kähler metrics

Abstract

Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a Kähler metric $\omega$ and a closed $(1,1)$-form $\alpha$. Assuming a uniform estimate for $\omega$, we prove higher order estimates and smooth convergence to a cscK metric coupled to a harmonic $(1,1)$-form. A simplification of the system is used to recover existence results for Kähler-Einstein metrics when $c_1(X)<0$. On Riemann surfaces with genus at least $2$, we show smooth convergence to the unique Kähler-Einstein metric from a large class of initial data.

14 pages