Global existence of solutions to parabolic Monge–Ampère equations on Riemannian manifolds

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation {ut−log{g−1det(gij+∇iju)}=−nloguinMn×(0,∞),u(x,0)=u0(x)inMn, where MnMn is a compact complete Riemannian manifold of dimension n≥2n≥2, gijgij denotes the metric of MnMn, g=det(gij)>0g=det(gij)>0 and u0(x)>1u0(x)>1 is a smooth function. We prove the global existence of solutions.