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

Abstract

We study the asymptotic behavior of the parabolic Monge-Ampère equation $\partial \phi (x,t)/\partial t=\text{log}(\text{det}(g(x)+\text{Hess}\phi (x,t))/\text{det}g(x))-\lambda \phi (x,t)$ in $𝕄\times (0,\infty )$, $\phi (x,0)={\phi }_0(x)$ in $𝕄$, where $𝕄$ is a compact complete Riemannian manifold, λ is a positive real parameter, and ${\phi }_0(x):𝕄\to ℝ$ is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.