Convergence of the Weak Kähler–Ricci Flow on Manifolds of General Type

Abstract

Abstract We study the Kähler–Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler–Ricci flow has long-time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler–Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge–Ampère flows in big classes that we develop, extending and refining the approach of Eyssidieux–Guedj–Zeriahi.