Coupled continuity equations for constant scalar curvature Kähler metrics

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 $ω$ and a closed $(1, 1)$-form $α$. Assuming a uniform estimate for $ω$, 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.

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