A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds

Abstract EN

We study the asymptotic behavior of the parabolic Monge-Ampère equation ∂φ(x,t)/∂t=log (det(g(x)+Hessφ(x,t))/detg(x))−λφ(x,t) in ?×(0,∞), φ(x,0)=φ0(x) in ?, where ? is a compact complete Riemannian manifold, λ is a positive real parameter, and φ0(x):?→ℝ is a smooth function.

We show a meaningful asymptotic result which is more general than those in Huisken, 1997.