Parabolic Monge–Ampère Equations on Riemannian Manifolds

On a compact Riemannian manifold (M, g) we consider the parabolic Monge–Ampère equation∂∂t ϕ(x, t)=log det(g(x)+Hess ϕ(x, t))det g(x)−λϕ(x, t)−f(x)ϕ(x, 0)=phiv;0(x).Hereλis a real parameter andf, ϕ0: M→R are smooth functions. We show existence ofϕfor all timestindependent ofλ. Ifλ>0, thenϕt=ϕ(·, t) converges exponentially towards a solutionϕ∞of the stationary problem ast→∞. In the special caseλ>0, f=0 one hasϕ∞=0 and we determine the convergence rateϕt→ϕ∞in theL2-norm more precisely.