Parabolic Monge–Ampère Equations on Riemannian Manifolds

AbstractOn a compact Riemannian manifold (M,g) we consider the parabolic Monge–Ampère equation∂∂tϕ(x,t)=logdet(g(x)+Hessϕ(x,t))detg(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