Coupled continuity equations for constant scalar curvature Kähler metrics

TL;DR

The paper analyzes a coupled elliptic system for a Kähler metric $\omega$ and a closed $(1,1)$-form $\alpha$, motivated by parabolic continuity methods for $cscK$ metrics. By reducing the system to a scalar Monge–Ampère formulation and establishing uniform $C^0$ bounds and higher‑order elliptic estimates, it proves smooth convergence to a $cscK$ metric paired with a harmonic $\alpha$ in the same class; in a simplified setting it recovers Kähler–Einstein existence when $c_1(X)<0$, and on Riemann surfaces of genus $\ge 2$ shows convergence from a broad class of initial data to the unique KE metric. The approach hinges on a priori estimates, cohomology class preservation, and a scalar reduction that yields complete regularity via elliptic theory. Together, these results extend the continuity/elliptic approach to canonical metrics by enabling convergence without constraining the coupling constant $\kappa$ and by handling both KE and cscK regimes in a unified framework.

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 $ω$ 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.

Theorems & Definitions (25)

• Theorem 1: Smooth convergence
• Remark 1
• Theorem 2: Smooth convergence to KE metric
• Theorem 3: Smooth convergence on Riemann surfaces
• Lemma 1: Li-Yuan-Zhang li2020new Proposition 2.3
• Lemma 2: Short-time existence
• proof
• Lemma 3: Stationary points
• proof
• Proposition 1: Scalar curvature lower bound