Curvature estimates for the level sets of solutions of the Monge-Amp\`{e}re equation $\det D^2 u=1$

Abstract

For the Monge-Amp\`{e}re equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation.Comment: This paper is part of the PhD thesis of the Third author Shujun Shi, and is submitted in Jan. 201